Theorem tfrlem2 2950

Description: Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 2949 into the main proof.

Assertion

Ref	Expression
tfrlem2	⊢ ((F Fn A ∧ G Fn A) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (A ∈ On → (∀w(A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))) → y = z))))

Distinct variable group(s):   w,A   w,B   w,F   w,G   x,w   y,w

Proof of Theorem tfrlem2

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . . . . . 13 ⊢ x ∈ V
2		visset 1350	. . . . . . . . . . . . 13 ⊢ y ∈ V
3	1, 2	fnop 2727	. . . . . . . . . . . 12 ⊢ ((F Fn A ∧ ⟨x, y⟩ ∈ F) → x ∈ A)
4	3	adantr 306	. . . . . . . . . . 11 ⊢ (((F Fn A ∧ ⟨x, y⟩ ∈ F) ∧ (G Fn A ∧ ⟨x, z⟩ ∈ G)) → x ∈ A)
5	4	an4s 390	. . . . . . . . . 10 ⊢ (((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → x ∈ A)
6		tfrlem1 2949	. . . . . . . . . . . 12 ⊢ (A ∈ On → ((F Fn A ∧ G Fn A) → (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) → ∀w ∈ A (F ‘w) = (G ‘w))))
7		fveq2 2832	. . . . . . . . . . . . . 14 ⊢ (w = x → (F ‘w) = (F ‘x))
8		fveq2 2832	. . . . . . . . . . . . . 14 ⊢ (w = x → (G ‘w) = (G ‘x))
9	7, 8	cleq12d 1115	. . . . . . . . . . . . 13 ⊢ (w = x → ((F ‘w) = (G ‘w) ↔ (F ‘x) = (G ‘x)))
10	9	rcla4v 1402	. . . . . . . . . . . 12 ⊢ (∀w ∈ A (F ‘w) = (G ‘w) → (x ∈ A → (F ‘x) = (G ‘x)))
11	6, 10	syl8 25	. . . . . . . . . . 11 ⊢ (A ∈ On → ((F Fn A ∧ G Fn A) → (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) → (x ∈ A → (F ‘x) = (G ‘x)))))
12	11	com4r 41	. . . . . . . . . 10 ⊢ (x ∈ A → (A ∈ On → ((F Fn A ∧ G Fn A) → (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) → (F ‘x) = (G ‘x)))))
13	5, 12	syl 12	. . . . . . . . 9 ⊢ (((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → (A ∈ On → ((F Fn A ∧ G Fn A) → (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) → (F ‘x) = (G ‘x)))))
14	13	exp 291	. . . . . . . 8 ⊢ ((F Fn A ∧ G Fn A) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (A ∈ On → ((F Fn A ∧ G Fn A) → (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) → (F ‘x) = (G ‘x))))))
15	14	com4r 41	. . . . . . 7 ⊢ ((F Fn A ∧ G Fn A) → ((F Fn A ∧ G Fn A) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (A ∈ On → (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) → (F ‘x) = (G ‘x))))))
16	15	pm2.43i 58	. . . . . 6 ⊢ ((F Fn A ∧ G Fn A) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (A ∈ On → (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) → (F ‘x) = (G ‘x)))))
17	16	imp43 288	. . . . 5 ⊢ ((((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) ∧ (A ∈ On ∧ ∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))) → (F ‘x) = (G ‘x))
18		df-ral 1205	. . . . . 6 ⊢ (∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))) ↔ ∀w(w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))
19	18	anbi2i 367	. . . . 5 ⊢ ((A ∈ On ∧ ∀w ∈ A ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))) ↔ (A ∈ On ∧ ∀w(w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))))
20	17, 19	sylan2br 348	. . . 4 ⊢ ((((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) ∧ (A ∈ On ∧ ∀w(w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))) → (F ‘x) = (G ‘x))
21	2	funfvopi 2853	. . . . . . . . . 10 ⊢ (Fun F → (⟨x, y⟩ ∈ F → (F ‘x) = y))
22	21	imp 277	. . . . . . . . 9 ⊢ ((Fun F ∧ ⟨x, y⟩ ∈ F) → (F ‘x) = y)
23		visset 1350	. . . . . . . . . . 11 ⊢ z ∈ V
24	23	funfvopi 2853	. . . . . . . . . 10 ⊢ (Fun G → (⟨x, z⟩ ∈ G → (G ‘x) = z))
25	24	imp 277	. . . . . . . . 9 ⊢ ((Fun G ∧ ⟨x, z⟩ ∈ G) → (G ‘x) = z)
26	22, 25	anim12i 268	. . . . . . . 8 ⊢ (((Fun F ∧ ⟨x, y⟩ ∈ F) ∧ (Fun G ∧ ⟨x, z⟩ ∈ G)) → ((F ‘x) = y ∧ (G ‘x) = z))
27	26	an4s 390	. . . . . . 7 ⊢ (((Fun F ∧ Fun G) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → ((F ‘x) = y ∧ (G ‘x) = z))
28		fnfun 2721	. . . . . . . 8 ⊢ (F Fn A → Fun F)
29		fnfun 2721	. . . . . . . 8 ⊢ (G Fn A → Fun G)
30	28, 29	anim12i 268	. . . . . . 7 ⊢ ((F Fn A ∧ G Fn A) → (Fun F ∧ Fun G))
31	27, 30	sylan 343	. . . . . 6 ⊢ (((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → ((F ‘x) = y ∧ (G ‘x) = z))
32		cleq12 1113	. . . . . 6 ⊢ (((F ‘x) = y ∧ (G ‘x) = z) → ((F ‘x) = (G ‘x) ↔ y = z))
33	31, 32	syl 12	. . . . 5 ⊢ (((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → ((F ‘x) = (G ‘x) ↔ y = z))
34	33	adantr 306	. . . 4 ⊢ ((((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) ∧ (A ∈ On ∧ ∀w(w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))) → ((F ‘x) = (G ‘x) ↔ y = z))
35	20, 34	mpbid 170	. . 3 ⊢ ((((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) ∧ (A ∈ On ∧ ∀w(w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))) → y = z)
36		abai 366	. . . . 5 ⊢ ((A ∈ On ∧ (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))) ↔ (A ∈ On ∧ (A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))))
37	36	bial 695	. . . 4 ⊢ (∀w(A ∈ On ∧ (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))) ↔ ∀w(A ∈ On ∧ (A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))))
38		19.28v 957	. . . 4 ⊢ (∀w(A ∈ On ∧ (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))) ↔ (A ∈ On ∧ ∀w(w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))))
39		19.28v 957	. . . 4 ⊢ (∀w(A ∈ On ∧ (A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))) ↔ (A ∈ On ∧ ∀w(A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))))
40	37, 38, 39	3bitr3r 157	. . 3 ⊢ ((A ∈ On ∧ ∀w(A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w)))))) ↔ (A ∈ On ∧ ∀w(w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))))
41	35, 40	sylan2b 347	. 2 ⊢ ((((F Fn A ∧ G Fn A) ∧ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) ∧ (A ∈ On ∧ ∀w(A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))))) → y = z)
42	41	exp43 301	1 ⊢ ((F Fn A ∧ G Fn A) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (A ∈ On → (∀w(A ∈ On → (w ∈ A → ((F ‘w) = (B ‘(F ↾ w)) ∧ (G ‘w) = (B ‘(G ↾ w))))) → y = z))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ⟨cop 1810  Oncon0 2199   ↾ cres 2412  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfrlem5 2953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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