Theorem isoini 2938

Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33.

Assertion

Ref	Expression
isoini	⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (H “ (A ∩ (◡R “ {D}))) = (B ∩ (◡S “ {(H ‘D)})))

Proof of Theorem isoini

Step	Hyp	Ref	Expression
1		isof1o 2931	. . . . . . . . 9 ⊢ (H Isom R, S (A, B) → H:A–1-1-onto→B)
2		f1ofo 2806	. . . . . . . . 9 ⊢ (H:A–1-1-onto→B → H:A–onto→B)
3		forn 2789	. . . . . . . . . 10 ⊢ (H:A–onto→B → ran H = B)
4	3	eleq2d 1156	. . . . . . . . 9 ⊢ (H:A–onto→B → (y ∈ ran H ↔ y ∈ B))
5	1, 2, 4	3syl 21	. . . . . . . 8 ⊢ (H Isom R, S (A, B) → (y ∈ ran H ↔ y ∈ B))
6		f1ofn 2801	. . . . . . . . 9 ⊢ (H:A–1-1-onto→B → H Fn A)
7		fvelrn 2883	. . . . . . . . 9 ⊢ (H Fn A → (y ∈ ran H ↔ ∃x ∈ A (H ‘x) = y))
8	1, 6, 7	3syl 21	. . . . . . . 8 ⊢ (H Isom R, S (A, B) → (y ∈ ran H ↔ ∃x ∈ A (H ‘x) = y))
9	5, 8	bitr3d 408	. . . . . . 7 ⊢ (H Isom R, S (A, B) → (y ∈ B ↔ ∃x ∈ A (H ‘x) = y))
10		fvex 2838	. . . . . . . . 9 ⊢ (H ‘D) ∈ V
11		visset 1350	. . . . . . . . . 10 ⊢ y ∈ V
12	11	eliniseg 2618	. . . . . . . . 9 ⊢ ((H ‘D) ∈ V → (y ∈ (◡S “ {(H ‘D)}) ↔ yS(H ‘D)))
13	10, 12	ax-mp 6	. . . . . . . 8 ⊢ (y ∈ (◡S “ {(H ‘D)}) ↔ yS(H ‘D))
14	13	a1i 7	. . . . . . 7 ⊢ (H Isom R, S (A, B) → (y ∈ (◡S “ {(H ‘D)}) ↔ yS(H ‘D)))
15	9, 14	anbi12d 476	. . . . . 6 ⊢ (H Isom R, S (A, B) → ((y ∈ B ∧ y ∈ (◡S “ {(H ‘D)})) ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D))))
16	15	adantr 306	. . . . 5 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((y ∈ B ∧ y ∈ (◡S “ {(H ‘D)})) ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D))))
17		visset 1350	. . . . . . . . . . . . . 14 ⊢ x ∈ V
18	17	eliniseg 2618	. . . . . . . . . . . . 13 ⊢ (D ∈ A → (x ∈ (◡R “ {D}) ↔ xRD))
19	18	anbi2d 468	. . . . . . . . . . . 12 ⊢ (D ∈ A → ((x ∈ A ∧ x ∈ (◡R “ {D})) ↔ (x ∈ A ∧ xRD)))
20		elin 1635	. . . . . . . . . . . 12 ⊢ (x ∈ (A ∩ (◡R “ {D})) ↔ (x ∈ A ∧ x ∈ (◡R “ {D})))
21	19, 20	syl5bb 410	. . . . . . . . . . 11 ⊢ (D ∈ A → (x ∈ (A ∩ (◡R “ {D})) ↔ (x ∈ A ∧ xRD)))
22	21	anbi1d 469	. . . . . . . . . 10 ⊢ (D ∈ A → ((x ∈ (A ∩ (◡R “ {D})) ∧ xHy) ↔ ((x ∈ A ∧ xRD) ∧ xHy)))
23		anass 336	. . . . . . . . . 10 ⊢ (((x ∈ A ∧ xRD) ∧ xHy) ↔ (x ∈ A ∧ (xRD ∧ xHy)))
24	22, 23	syl6bb 414	. . . . . . . . 9 ⊢ (D ∈ A → ((x ∈ (A ∩ (◡R “ {D})) ∧ xHy) ↔ (x ∈ A ∧ (xRD ∧ xHy))))
25	24	adantl 305	. . . . . . . 8 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((x ∈ (A ∩ (◡R “ {D})) ∧ xHy) ↔ (x ∈ A ∧ (xRD ∧ xHy))))
26		isorel 2932	. . . . . . . . . . . . . 14 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → (xRD ↔ (H ‘x)S(H ‘D)))
27	11	fnfvbr 2855	. . . . . . . . . . . . . . . . 17 ⊢ ((H Fn A ∧ x ∈ A) → ((H ‘x) = y ↔ xHy))
28	27	bicomd 399	. . . . . . . . . . . . . . . 16 ⊢ ((H Fn A ∧ x ∈ A) → (xHy ↔ (H ‘x) = y))
29	1, 6	syl 12	. . . . . . . . . . . . . . . 16 ⊢ (H Isom R, S (A, B) → H Fn A)
30	28, 29	sylan 343	. . . . . . . . . . . . . . 15 ⊢ ((H Isom R, S (A, B) ∧ x ∈ A) → (xHy ↔ (H ‘x) = y))
31	30	adantrr 312	. . . . . . . . . . . . . 14 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → (xHy ↔ (H ‘x) = y))
32	26, 31	anbi12d 476	. . . . . . . . . . . . 13 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → ((xRD ∧ xHy) ↔ ((H ‘x)S(H ‘D) ∧ (H ‘x) = y)))
33		ancom 333	. . . . . . . . . . . . . 14 ⊢ (((H ‘x)S(H ‘D) ∧ (H ‘x) = y) ↔ ((H ‘x) = y ∧ (H ‘x)S(H ‘D)))
34		breq1 2065	. . . . . . . . . . . . . . 15 ⊢ ((H ‘x) = y → ((H ‘x)S(H ‘D) ↔ yS(H ‘D)))
35	34	pm5.32i 489	. . . . . . . . . . . . . 14 ⊢ (((H ‘x) = y ∧ (H ‘x)S(H ‘D)) ↔ ((H ‘x) = y ∧ yS(H ‘D)))
36	33, 35	bitr 151	. . . . . . . . . . . . 13 ⊢ (((H ‘x)S(H ‘D) ∧ (H ‘x) = y) ↔ ((H ‘x) = y ∧ yS(H ‘D)))
37	32, 36	syl6bb 414	. . . . . . . . . . . 12 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D))))
38	37	exp32 294	. . . . . . . . . . 11 ⊢ (H Isom R, S (A, B) → (x ∈ A → (D ∈ A → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D))))))
39	38	com23 32	. . . . . . . . . 10 ⊢ (H Isom R, S (A, B) → (D ∈ A → (x ∈ A → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D))))))
40	39	imp 277	. . . . . . . . 9 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (x ∈ A → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D)))))
41	40	pm5.32d 491	. . . . . . . 8 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((x ∈ A ∧ (xRD ∧ xHy)) ↔ (x ∈ A ∧ ((H ‘x) = y ∧ yS(H ‘D)))))
42	25, 41	bitrd 406	. . . . . . 7 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((x ∈ (A ∩ (◡R “ {D})) ∧ xHy) ↔ (x ∈ A ∧ ((H ‘x) = y ∧ yS(H ‘D)))))
43	42	birexdv2 1222	. . . . . 6 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (∃x ∈ (A ∩ (◡R “ {D}))xHy ↔ ∃x ∈ A ((H ‘x) = y ∧ yS(H ‘D))))
44		r19.41v 1302	. . . . . 6 ⊢ (∃x ∈ A ((H ‘x) = y ∧ yS(H ‘D)) ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D)))
45	43, 44	syl6bb 414	. . . . 5 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (∃x ∈ (A ∩ (◡R “ {D}))xHy ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D))))
46	16, 45	bitr4d 409	. . . 4 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((y ∈ B ∧ y ∈ (◡S “ {(H ‘D)})) ↔ ∃x ∈ (A ∩ (◡R “ {D}))xHy))
47		elin 1635	. . . 4 ⊢ (y ∈ (B ∩ (◡S “ {(H ‘D)})) ↔ (y ∈ B ∧ y ∈ (◡S “ {(H ‘D)})))
48	46, 47	syl5bb 410	. . 3 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (y ∈ (B ∩ (◡S “ {(H ‘D)})) ↔ ∃x ∈ (A ∩ (◡R “ {D}))xHy))
49	48	biabrdv 1184	. 2 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (B ∩ (◡S “ {(H ‘D)})) = {y∣∃x ∈ (A ∩ (◡R “ {D}))xHy})
50		dfima2 2604	. 2 ⊢ (H “ (A ∩ (◡R “ {D}))) = {y∣∃x ∈ (A ∩ (◡R “ {D}))xHy}
51	49, 50	syl6reqr 1143	1 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (H “ (A ∩ (◡R “ {D}))) = (B ∩ (◡S “ {(H ‘D)})))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ∩ cin 1486  {csn 1808   class class class wbr 2054  ^◡ccnv 2409  ran crn 2411   “ cima 2413   Fn wfn 2417  –onto→wfo 2420  –1-1-onto→wf1o 2421   ‘cfv 2422   Isom wiso 2423

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-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-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-br 2063  df-opab 2098  df-id 2125  df-xp 2424  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-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439

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