Theorem rdglem2 2976

Description: Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use.

Assertion

Ref	Expression
rdglem2	⊢ {⟨x, y⟩∣((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))} = {⟨z, y⟩∣((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z))}

Distinct variable group(s):   x,y,z   x,A,z   x,H,z

Proof of Theorem rdglem2

Step	Hyp	Ref	Expression
1		opeq1 1876	. . . . . . 7 ⊢ (x = z → ⟨x, y⟩ = ⟨z, y⟩)
2	1	cleq2d 1112	. . . . . 6 ⊢ (x = z → (w = ⟨x, y⟩ ↔ w = ⟨z, y⟩))
3		cleq1 1107	. . . . . . . 8 ⊢ (x = z → (x = ∅ ↔ z = ∅))
4	3	anbi1d 469	. . . . . . 7 ⊢ (x = z → ((x = ∅ ∧ y = A) ↔ (z = ∅ ∧ y = A)))
5		dmeq 2531	. . . . . . . . . . 11 ⊢ (x = z → dom x = dom z)
6		limeq 2211	. . . . . . . . . . 11 ⊢ (dom x = dom z → (Lim dom x ↔ Lim dom z))
7	5, 6	syl 12	. . . . . . . . . 10 ⊢ (x = z → (Lim dom x ↔ Lim dom z))
8	3, 7	orbi12d 475	. . . . . . . . 9 ⊢ (x = z → ((x = ∅ ∨ Lim dom x) ↔ (z = ∅ ∨ Lim dom z)))
9	8	negbid 463	. . . . . . . 8 ⊢ (x = z → (¬ (x = ∅ ∨ Lim dom x) ↔ ¬ (z = ∅ ∨ Lim dom z)))
10		unieq 1927	. . . . . . . . . . . 12 ⊢ (dom x = dom z → ∪dom x = ∪dom z)
11		fveq2 2832	. . . . . . . . . . . 12 ⊢ (∪dom x = ∪dom z → (x ‘∪dom x) = (x ‘∪dom z))
12	5, 10, 11	3syl 21	. . . . . . . . . . 11 ⊢ (x = z → (x ‘∪dom x) = (x ‘∪dom z))
13		fveq1 2831	. . . . . . . . . . 11 ⊢ (x = z → (x ‘∪dom z) = (z ‘∪dom z))
14	12, 13	eqtrd 1128	. . . . . . . . . 10 ⊢ (x = z → (x ‘∪dom x) = (z ‘∪dom z))
15	14	fveq2d 2836	. . . . . . . . 9 ⊢ (x = z → (H ‘(x ‘∪dom x)) = (H ‘(z ‘∪dom z)))
16	15	cleq2d 1112	. . . . . . . 8 ⊢ (x = z → (y = (H ‘(x ‘∪dom x)) ↔ y = (H ‘(z ‘∪dom z))))
17	9, 16	anbi12d 476	. . . . . . 7 ⊢ (x = z → ((¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ↔ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z)))))
18		rneq 2555	. . . . . . . . . 10 ⊢ (x = z → ran x = ran z)
19	18	unieqd 1929	. . . . . . . . 9 ⊢ (x = z → ∪ran x = ∪ran z)
20	19	cleq2d 1112	. . . . . . . 8 ⊢ (x = z → (y = ∪ran x ↔ y = ∪ran z))
21	7, 20	anbi12d 476	. . . . . . 7 ⊢ (x = z → ((Lim dom x ∧ y = ∪ran x) ↔ (Lim dom z ∧ y = ∪ran z)))
22	4, 17, 21	bi3ord 635	. . . . . 6 ⊢ (x = z → (((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x)) ↔ ((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z))))
23	2, 22	anbi12d 476	. . . . 5 ⊢ (x = z → ((w = ⟨x, y⟩ ∧ ((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))) ↔ (w = ⟨z, y⟩ ∧ ((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z)))))
24	23	biexdv 936	. . . 4 ⊢ (x = z → (∃y(w = ⟨x, y⟩ ∧ ((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))) ↔ ∃y(w = ⟨z, y⟩ ∧ ((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z)))))
25	24	cbvexv 973	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ ((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))) ↔ ∃z∃y(w = ⟨z, y⟩ ∧ ((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z))))
26	25	biabi 1181	. 2 ⊢ {w∣∃x∃y(w = ⟨x, y⟩ ∧ ((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x)))} = {w∣∃z∃y(w = ⟨z, y⟩ ∧ ((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z)))}
27		df-opab 2098	. 2 ⊢ {⟨x, y⟩∣((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))} = {w∣∃x∃y(w = ⟨x, y⟩ ∧ ((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x)))}
28		df-opab 2098	. 2 ⊢ {⟨z, y⟩∣((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z))} = {w∣∃z∃y(w = ⟨z, y⟩ ∧ ((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z)))}
29	26, 27, 28	3eqtr4 1126	1 ⊢ {⟨x, y⟩∣((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))} = {⟨z, y⟩∣((z = ∅ ∧ y = A) ∨ (¬ (z = ∅ ∨ Lim dom z) ∧ y = (H ‘(z ‘∪dom z))) ∨ (Lim dom z ∧ y = ∪ran z))}

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∨ w3o 580  ∃wex 678   = weq 797  {cab 1090   = wceq 1091  ∅c0 1707  ⟨cop 1810  ∪cuni 1919  {copab 2055  Lim wlim 2200  dom cdm 2410  ran crn 2411   ‘cfv 2422

This theorem is referenced by:  rdgval 2978  rdgzer 2979  rdgsuc 2980  rdglim 2981

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-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-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  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-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-lim 2204  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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