Theorem isoini 5498

Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 20-Apr-2004.)

Assertion

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

Proof of Theorem isoini

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3220	. . . . 5 ⊢ (y ∈ (B ∩ (◡S “ {(H ‘D)})) ↔ (y ∈ B ∧ y ∈ (◡S “ {(H ‘D)})))
2		eliniseg 5021	. . . . . 6 ⊢ (y ∈ (◡S “ {(H ‘D)}) ↔ yS(H ‘D))
3	2	anbi2i 675	. . . . 5 ⊢ ((y ∈ B ∧ y ∈ (◡S “ {(H ‘D)})) ↔ (y ∈ B ∧ yS(H ‘D)))
4	1, 3	bitri 240	. . . 4 ⊢ (y ∈ (B ∩ (◡S “ {(H ‘D)})) ↔ (y ∈ B ∧ yS(H ‘D)))
5		isof1o 5489	. . . . . . . . . . 11 ⊢ (H Isom R, S (A, B) → H:A–1-1-onto→B)
6		f1ofo 5294	. . . . . . . . . . 11 ⊢ (H:A–1-1-onto→B → H:A–onto→B)
7	5, 6	syl 15	. . . . . . . . . 10 ⊢ (H Isom R, S (A, B) → H:A–onto→B)
8		forn 5273	. . . . . . . . . 10 ⊢ (H:A–onto→B → ran H = B)
9	7, 8	syl 15	. . . . . . . . 9 ⊢ (H Isom R, S (A, B) → ran H = B)
10	9	eleq2d 2420	. . . . . . . 8 ⊢ (H Isom R, S (A, B) → (y ∈ ran H ↔ y ∈ B))
11		f1ofn 5289	. . . . . . . . . 10 ⊢ (H:A–1-1-onto→B → H Fn A)
12	5, 11	syl 15	. . . . . . . . 9 ⊢ (H Isom R, S (A, B) → H Fn A)
13		fvelrnb 5366	. . . . . . . . 9 ⊢ (H Fn A → (y ∈ ran H ↔ ∃x ∈ A (H ‘x) = y))
14	12, 13	syl 15	. . . . . . . 8 ⊢ (H Isom R, S (A, B) → (y ∈ ran H ↔ ∃x ∈ A (H ‘x) = y))
15	10, 14	bitr3d 246	. . . . . . 7 ⊢ (H Isom R, S (A, B) → (y ∈ B ↔ ∃x ∈ A (H ‘x) = y))
16	15	anbi1d 685	. . . . . 6 ⊢ (H Isom R, S (A, B) → ((y ∈ B ∧ yS(H ‘D)) ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D))))
17	16	adantr 451	. . . . 5 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((y ∈ B ∧ yS(H ‘D)) ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D))))
18		elin 3220	. . . . . . . . . . 11 ⊢ (x ∈ (A ∩ (◡R “ {D})) ↔ (x ∈ A ∧ x ∈ (◡R “ {D})))
19		eliniseg 5021	. . . . . . . . . . . 12 ⊢ (x ∈ (◡R “ {D}) ↔ xRD)
20	19	anbi2i 675	. . . . . . . . . . 11 ⊢ ((x ∈ A ∧ x ∈ (◡R “ {D})) ↔ (x ∈ A ∧ xRD))
21	18, 20	bitri 240	. . . . . . . . . 10 ⊢ (x ∈ (A ∩ (◡R “ {D})) ↔ (x ∈ A ∧ xRD))
22	21	anbi1i 676	. . . . . . . . 9 ⊢ ((x ∈ (A ∩ (◡R “ {D})) ∧ xHy) ↔ ((x ∈ A ∧ xRD) ∧ xHy))
23		anass 630	. . . . . . . . 9 ⊢ (((x ∈ A ∧ xRD) ∧ xHy) ↔ (x ∈ A ∧ (xRD ∧ xHy)))
24	22, 23	bitri 240	. . . . . . . 8 ⊢ ((x ∈ (A ∩ (◡R “ {D})) ∧ xHy) ↔ (x ∈ A ∧ (xRD ∧ xHy)))
25		fnbrfvb 5359	. . . . . . . . . . . . . . . . 17 ⊢ ((H Fn A ∧ x ∈ A) → ((H ‘x) = y ↔ xHy))
26	12, 25	sylan 457	. . . . . . . . . . . . . . . 16 ⊢ ((H Isom R, S (A, B) ∧ x ∈ A) → ((H ‘x) = y ↔ xHy))
27	26	adantrr 697	. . . . . . . . . . . . . . 15 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → ((H ‘x) = y ↔ xHy))
28	27	bicomd 192	. . . . . . . . . . . . . 14 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → (xHy ↔ (H ‘x) = y))
29		isorel 5490	. . . . . . . . . . . . . 14 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → (xRD ↔ (H ‘x)S(H ‘D)))
30	28, 29	anbi12d 691	. . . . . . . . . . . . 13 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → ((xHy ∧ xRD) ↔ ((H ‘x) = y ∧ (H ‘x)S(H ‘D))))
31		ancom 437	. . . . . . . . . . . . 13 ⊢ ((xHy ∧ xRD) ↔ (xRD ∧ xHy))
32		breq1 4643	. . . . . . . . . . . . . 14 ⊢ ((H ‘x) = y → ((H ‘x)S(H ‘D) ↔ yS(H ‘D)))
33	32	pm5.32i 618	. . . . . . . . . . . . 13 ⊢ (((H ‘x) = y ∧ (H ‘x)S(H ‘D)) ↔ ((H ‘x) = y ∧ yS(H ‘D)))
34	30, 31, 33	3bitr3g 278	. . . . . . . . . . . 12 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ D ∈ A)) → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D))))
35	34	exp32 588	. . . . . . . . . . 11 ⊢ (H Isom R, S (A, B) → (x ∈ A → (D ∈ A → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D))))))
36	35	com23 72	. . . . . . . . . 10 ⊢ (H Isom R, S (A, B) → (D ∈ A → (x ∈ A → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D))))))
37	36	imp 418	. . . . . . . . 9 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (x ∈ A → ((xRD ∧ xHy) ↔ ((H ‘x) = y ∧ yS(H ‘D)))))
38	37	pm5.32d 620	. . . . . . . 8 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((x ∈ A ∧ (xRD ∧ xHy)) ↔ (x ∈ A ∧ ((H ‘x) = y ∧ yS(H ‘D)))))
39	24, 38	syl5bb 248	. . . . . . 7 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((x ∈ (A ∩ (◡R “ {D})) ∧ xHy) ↔ (x ∈ A ∧ ((H ‘x) = y ∧ yS(H ‘D)))))
40	39	rexbidv2 2638	. . . . . 6 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (∃x ∈ (A ∩ (◡R “ {D}))xHy ↔ ∃x ∈ A ((H ‘x) = y ∧ yS(H ‘D))))
41		r19.41v 2765	. . . . . 6 ⊢ (∃x ∈ A ((H ‘x) = y ∧ yS(H ‘D)) ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D)))
42	40, 41	syl6bb 252	. . . . 5 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (∃x ∈ (A ∩ (◡R “ {D}))xHy ↔ (∃x ∈ A (H ‘x) = y ∧ yS(H ‘D))))
43	17, 42	bitr4d 247	. . . 4 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → ((y ∈ B ∧ yS(H ‘D)) ↔ ∃x ∈ (A ∩ (◡R “ {D}))xHy))
44	4, 43	syl5bb 248	. . 3 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (y ∈ (B ∩ (◡S “ {(H ‘D)})) ↔ ∃x ∈ (A ∩ (◡R “ {D}))xHy))
45	44	abbi2dv 2469	. 2 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (B ∩ (◡S “ {(H ‘D)})) = {y ∣ ∃x ∈ (A ∩ (◡R “ {D}))xHy})
46		df-ima 4728	. 2 ⊢ (H “ (A ∩ (◡R “ {D}))) = {y ∣ ∃x ∈ (A ∩ (◡R “ {D}))xHy}
47	45, 46	syl6reqr 2404	1 ⊢ ((H Isom R, S (A, B) ∧ D ∈ A) → (H “ (A ∩ (◡R “ {D}))) = (B ∩ (◡S “ {(H ‘D)})))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616   ∩ cin 3209  {csn 3738   class class class wbr 4640   “ cima 4723  ^◡ccnv 4772  ran crn 4774   Fn wfn 4777  –onto→wfo 4780  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is referenced by:  isoini2  5499