Theorem f1oiso2 5501

Description: Any one-to-one onto function determines an isomorphism with an induced relation S. (Contributed by Mario Carneiro, 9-Mar-2013.)

Hypothesis

Ref	Expression
f1oiso2.1	⊢ S = {⟨x, y⟩ ∣ ((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y))}

Assertion

Ref	Expression
f1oiso2	⊢ (H:A–1-1-onto→B → H Isom R, S (A, B))

Distinct variable groups:   x,A,y   x,B,y   x,H,y   x,R,y

Allowed substitution hints:   S(x,y)

Proof of Theorem f1oiso2

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oiso2.1	. . 3 ⊢ S = {⟨x, y⟩ ∣ ((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y))}
2		f1ocnvdm 5482	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ x ∈ B) → (◡H ‘x) ∈ A)
3	2	adantrr 697	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B)) → (◡H ‘x) ∈ A)
4	3	3adant3 975	. . . . . . 7 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) → (◡H ‘x) ∈ A)
5		f1ocnvdm 5482	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ y ∈ B) → (◡H ‘y) ∈ A)
6	5	adantrl 696	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B)) → (◡H ‘y) ∈ A)
7	6	3adant3 975	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) → (◡H ‘y) ∈ A)
8		f1ocnvfv2 5478	. . . . . . . . . . 11 ⊢ ((H:A–1-1-onto→B ∧ x ∈ B) → (H ‘(◡H ‘x)) = x)
9	8	eqcomd 2358	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ x ∈ B) → x = (H ‘(◡H ‘x)))
10		f1ocnvfv2 5478	. . . . . . . . . . 11 ⊢ ((H:A–1-1-onto→B ∧ y ∈ B) → (H ‘(◡H ‘y)) = y)
11	10	eqcomd 2358	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ y ∈ B) → y = (H ‘(◡H ‘y)))
12	9, 11	anim12dan 810	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B)) → (x = (H ‘(◡H ‘x)) ∧ y = (H ‘(◡H ‘y))))
13	12	3adant3 975	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) → (x = (H ‘(◡H ‘x)) ∧ y = (H ‘(◡H ‘y))))
14		simp3 957	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) → (◡H ‘x)R(◡H ‘y))
15		fveq2 5329	. . . . . . . . . . . 12 ⊢ (w = (◡H ‘y) → (H ‘w) = (H ‘(◡H ‘y)))
16	15	eqeq2d 2364	. . . . . . . . . . 11 ⊢ (w = (◡H ‘y) → (y = (H ‘w) ↔ y = (H ‘(◡H ‘y))))
17	16	anbi2d 684	. . . . . . . . . 10 ⊢ (w = (◡H ‘y) → ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘w)) ↔ (x = (H ‘(◡H ‘x)) ∧ y = (H ‘(◡H ‘y)))))
18		breq2 4644	. . . . . . . . . 10 ⊢ (w = (◡H ‘y) → ((◡H ‘x)Rw ↔ (◡H ‘x)R(◡H ‘y)))
19	17, 18	anbi12d 691	. . . . . . . . 9 ⊢ (w = (◡H ‘y) → (((x = (H ‘(◡H ‘x)) ∧ y = (H ‘w)) ∧ (◡H ‘x)Rw) ↔ ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘(◡H ‘y))) ∧ (◡H ‘x)R(◡H ‘y))))
20	19	rspcev 2956	. . . . . . . 8 ⊢ (((◡H ‘y) ∈ A ∧ ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘(◡H ‘y))) ∧ (◡H ‘x)R(◡H ‘y))) → ∃w ∈ A ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘w)) ∧ (◡H ‘x)Rw))
21	7, 13, 14, 20	syl12anc 1180	. . . . . . 7 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) → ∃w ∈ A ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘w)) ∧ (◡H ‘x)Rw))
22		fveq2 5329	. . . . . . . . . . . 12 ⊢ (z = (◡H ‘x) → (H ‘z) = (H ‘(◡H ‘x)))
23	22	eqeq2d 2364	. . . . . . . . . . 11 ⊢ (z = (◡H ‘x) → (x = (H ‘z) ↔ x = (H ‘(◡H ‘x))))
24	23	anbi1d 685	. . . . . . . . . 10 ⊢ (z = (◡H ‘x) → ((x = (H ‘z) ∧ y = (H ‘w)) ↔ (x = (H ‘(◡H ‘x)) ∧ y = (H ‘w))))
25		breq1 4643	. . . . . . . . . 10 ⊢ (z = (◡H ‘x) → (zRw ↔ (◡H ‘x)Rw))
26	24, 25	anbi12d 691	. . . . . . . . 9 ⊢ (z = (◡H ‘x) → (((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw) ↔ ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘w)) ∧ (◡H ‘x)Rw)))
27	26	rexbidv 2636	. . . . . . . 8 ⊢ (z = (◡H ‘x) → (∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw) ↔ ∃w ∈ A ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘w)) ∧ (◡H ‘x)Rw)))
28	27	rspcev 2956	. . . . . . 7 ⊢ (((◡H ‘x) ∈ A ∧ ∃w ∈ A ((x = (H ‘(◡H ‘x)) ∧ y = (H ‘w)) ∧ (◡H ‘x)Rw)) → ∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw))
29	4, 21, 28	syl2anc 642	. . . . . 6 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) → ∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw))
30	29	3expib 1154	. . . . 5 ⊢ (H:A–1-1-onto→B → (((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) → ∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)))
31		simp3ll 1026	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → x = (H ‘z))
32		simp1 955	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → H:A–1-1-onto→B)
33		simp2l 981	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → z ∈ A)
34		f1of 5288	. . . . . . . . . . 11 ⊢ (H:A–1-1-onto→B → H:A–→B)
35		ffvelrn 5416	. . . . . . . . . . 11 ⊢ ((H:A–→B ∧ z ∈ A) → (H ‘z) ∈ B)
36	34, 35	sylan 457	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ z ∈ A) → (H ‘z) ∈ B)
37	32, 33, 36	syl2anc 642	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → (H ‘z) ∈ B)
38	31, 37	eqeltrd 2427	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → x ∈ B)
39		simp3lr 1027	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → y = (H ‘w))
40		simp2r 982	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → w ∈ A)
41		ffvelrn 5416	. . . . . . . . . . 11 ⊢ ((H:A–→B ∧ w ∈ A) → (H ‘w) ∈ B)
42	34, 41	sylan 457	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ w ∈ A) → (H ‘w) ∈ B)
43	32, 40, 42	syl2anc 642	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → (H ‘w) ∈ B)
44	39, 43	eqeltrd 2427	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → y ∈ B)
45		simp3r 984	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → zRw)
46	31	eqcomd 2358	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → (H ‘z) = x)
47		f1ocnvfv 5479	. . . . . . . . . . 11 ⊢ ((H:A–1-1-onto→B ∧ z ∈ A) → ((H ‘z) = x → (◡H ‘x) = z))
48	32, 33, 47	syl2anc 642	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → ((H ‘z) = x → (◡H ‘x) = z))
49	46, 48	mpd 14	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → (◡H ‘x) = z)
50	39	eqcomd 2358	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → (H ‘w) = y)
51		f1ocnvfv 5479	. . . . . . . . . . 11 ⊢ ((H:A–1-1-onto→B ∧ w ∈ A) → ((H ‘w) = y → (◡H ‘y) = w))
52	32, 40, 51	syl2anc 642	. . . . . . . . . 10 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → ((H ‘w) = y → (◡H ‘y) = w))
53	50, 52	mpd 14	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → (◡H ‘y) = w)
54	45, 49, 53	3brtr4d 4670	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → (◡H ‘x)R(◡H ‘y))
55	38, 44, 54	jca31 520	. . . . . . 7 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ A ∧ w ∈ A) ∧ ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)) → ((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)))
56	55	3exp 1150	. . . . . 6 ⊢ (H:A–1-1-onto→B → ((z ∈ A ∧ w ∈ A) → (((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw) → ((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)))))
57	56	rexlimdvv 2745	. . . . 5 ⊢ (H:A–1-1-onto→B → (∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw) → ((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y))))
58	30, 57	impbid 183	. . . 4 ⊢ (H:A–1-1-onto→B → (((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y)) ↔ ∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)))
59	58	opabbidv 4626	. . 3 ⊢ (H:A–1-1-onto→B → {⟨x, y⟩ ∣ ((x ∈ B ∧ y ∈ B) ∧ (◡H ‘x)R(◡H ‘y))} = {⟨x, y⟩ ∣ ∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)})
60	1, 59	syl5eq 2397	. 2 ⊢ (H:A–1-1-onto→B → S = {⟨x, y⟩ ∣ ∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)})
61		f1oiso 5500	. 2 ⊢ ((H:A–1-1-onto→B ∧ S = {⟨x, y⟩ ∣ ∃z ∈ A ∃w ∈ A ((x = (H ‘z) ∧ y = (H ‘w)) ∧ zRw)}) → H Isom R, S (A, B))
62	60, 61	mpdan 649	1 ⊢ (H:A–1-1-onto→B → H Isom R, S (A, B))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  {copab 4623   class class class wbr 4640  ^◡ccnv 4772  –→wf 4778  –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: (None)