Theorem f1oiso 5500

Description: Any one-to-one onto function determines an isomorphism with an induced relation S. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by set.mm contributors, 30-Apr-2004.)

Assertion

Ref	Expression
f1oiso	⊢ ((H:A–1-1-onto→B ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) → H Isom R, S (A, B))

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

Allowed substitution hints:   B(z, w)   S(x, y, z, w)

Proof of Theorem f1oiso

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 443	. 2 ⊢ ((H:A–1-1-onto→B ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) → H:A–1-1-onto→B)
2		f1of1 5287	. . 3 ⊢ (H:A–1-1-onto→B → H:A–1-1→B)
3		df-br 4641	. . . . 5 ⊢ ((H ‘v)S(H ‘u) ↔ ⟨(H ‘v), (H ‘u)⟩ ∈ S)
4		eleq2 2414	. . . . . . 7 ⊢ (S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)} → (⟨(H ‘v), (H ‘u)⟩ ∈ S ↔ ⟨(H ‘v), (H ‘u)⟩ ∈ {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}))
5		fvex 5340	. . . . . . . . 9 ⊢ (H ‘v) ∈ V
6		fvex 5340	. . . . . . . . 9 ⊢ (H ‘u) ∈ V
7		eqeq1 2359	. . . . . . . . . . . 12 ⊢ (z = (H ‘v) → (z = (H ‘x) ↔ (H ‘v) = (H ‘x)))
8	7	anbi1d 685	. . . . . . . . . . 11 ⊢ (z = (H ‘v) → ((z = (H ‘x) ∧ w = (H ‘y)) ↔ ((H ‘v) = (H ‘x) ∧ w = (H ‘y))))
9	8	anbi1d 685	. . . . . . . . . 10 ⊢ (z = (H ‘v) → (((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy) ↔ (((H ‘v) = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)))
10	9	2rexbidv 2658	. . . . . . . . 9 ⊢ (z = (H ‘v) → (∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy) ↔ ∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)))
11		eqeq1 2359	. . . . . . . . . . . 12 ⊢ (w = (H ‘u) → (w = (H ‘y) ↔ (H ‘u) = (H ‘y)))
12	11	anbi2d 684	. . . . . . . . . . 11 ⊢ (w = (H ‘u) → (((H ‘v) = (H ‘x) ∧ w = (H ‘y)) ↔ ((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y))))
13	12	anbi1d 685	. . . . . . . . . 10 ⊢ (w = (H ‘u) → ((((H ‘v) = (H ‘x) ∧ w = (H ‘y)) ∧ xRy) ↔ (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy)))
14	13	2rexbidv 2658	. . . . . . . . 9 ⊢ (w = (H ‘u) → (∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ w = (H ‘y)) ∧ xRy) ↔ ∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy)))
15	5, 6, 10, 14	opelopab 4709	. . . . . . . 8 ⊢ (⟨(H ‘v), (H ‘u)⟩ ∈ {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)} ↔ ∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy))
16		anass 630	. . . . . . . . . . . . . . 15 ⊢ ((((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ ((H ‘v) = (H ‘x) ∧ ((H ‘u) = (H ‘y) ∧ xRy)))
17		f1fveq 5474	. . . . . . . . . . . . . . . . . 18 ⊢ ((H:A–1-1→B ∧ (v ∈ A ∧ x ∈ A)) → ((H ‘v) = (H ‘x) ↔ v = x))
18		eqcom 2355	. . . . . . . . . . . . . . . . . 18 ⊢ (v = x ↔ x = v)
19	17, 18	syl6bb 252	. . . . . . . . . . . . . . . . 17 ⊢ ((H:A–1-1→B ∧ (v ∈ A ∧ x ∈ A)) → ((H ‘v) = (H ‘x) ↔ x = v))
20	19	anassrs 629	. . . . . . . . . . . . . . . 16 ⊢ (((H:A–1-1→B ∧ v ∈ A) ∧ x ∈ A) → ((H ‘v) = (H ‘x) ↔ x = v))
21	20	anbi1d 685	. . . . . . . . . . . . . . 15 ⊢ (((H:A–1-1→B ∧ v ∈ A) ∧ x ∈ A) → (((H ‘v) = (H ‘x) ∧ ((H ‘u) = (H ‘y) ∧ xRy)) ↔ (x = v ∧ ((H ‘u) = (H ‘y) ∧ xRy))))
22	16, 21	syl5bb 248	. . . . . . . . . . . . . 14 ⊢ (((H:A–1-1→B ∧ v ∈ A) ∧ x ∈ A) → ((((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ (x = v ∧ ((H ‘u) = (H ‘y) ∧ xRy))))
23	22	rexbidv 2636	. . . . . . . . . . . . 13 ⊢ (((H:A–1-1→B ∧ v ∈ A) ∧ x ∈ A) → (∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ ∃y ∈ A (x = v ∧ ((H ‘u) = (H ‘y) ∧ xRy))))
24		r19.42v 2766	. . . . . . . . . . . . 13 ⊢ (∃y ∈ A (x = v ∧ ((H ‘u) = (H ‘y) ∧ xRy)) ↔ (x = v ∧ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ xRy)))
25	23, 24	syl6bb 252	. . . . . . . . . . . 12 ⊢ (((H:A–1-1→B ∧ v ∈ A) ∧ x ∈ A) → (∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ (x = v ∧ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ xRy))))
26	25	rexbidva 2632	. . . . . . . . . . 11 ⊢ ((H:A–1-1→B ∧ v ∈ A) → (∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ ∃x ∈ A (x = v ∧ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ xRy))))
27		breq1 4643	. . . . . . . . . . . . . . 15 ⊢ (x = v → (xRy ↔ vRy))
28	27	anbi2d 684	. . . . . . . . . . . . . 14 ⊢ (x = v → (((H ‘u) = (H ‘y) ∧ xRy) ↔ ((H ‘u) = (H ‘y) ∧ vRy)))
29	28	rexbidv 2636	. . . . . . . . . . . . 13 ⊢ (x = v → (∃y ∈ A ((H ‘u) = (H ‘y) ∧ xRy) ↔ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ vRy)))
30	29	ceqsrexv 2973	. . . . . . . . . . . 12 ⊢ (v ∈ A → (∃x ∈ A (x = v ∧ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ xRy)) ↔ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ vRy)))
31	30	adantl 452	. . . . . . . . . . 11 ⊢ ((H:A–1-1→B ∧ v ∈ A) → (∃x ∈ A (x = v ∧ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ xRy)) ↔ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ vRy)))
32	26, 31	bitrd 244	. . . . . . . . . 10 ⊢ ((H:A–1-1→B ∧ v ∈ A) → (∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ ∃y ∈ A ((H ‘u) = (H ‘y) ∧ vRy)))
33		f1fveq 5474	. . . . . . . . . . . . . . 15 ⊢ ((H:A–1-1→B ∧ (u ∈ A ∧ y ∈ A)) → ((H ‘u) = (H ‘y) ↔ u = y))
34		eqcom 2355	. . . . . . . . . . . . . . 15 ⊢ (u = y ↔ y = u)
35	33, 34	syl6bb 252	. . . . . . . . . . . . . 14 ⊢ ((H:A–1-1→B ∧ (u ∈ A ∧ y ∈ A)) → ((H ‘u) = (H ‘y) ↔ y = u))
36	35	anassrs 629	. . . . . . . . . . . . 13 ⊢ (((H:A–1-1→B ∧ u ∈ A) ∧ y ∈ A) → ((H ‘u) = (H ‘y) ↔ y = u))
37	36	anbi1d 685	. . . . . . . . . . . 12 ⊢ (((H:A–1-1→B ∧ u ∈ A) ∧ y ∈ A) → (((H ‘u) = (H ‘y) ∧ vRy) ↔ (y = u ∧ vRy)))
38	37	rexbidva 2632	. . . . . . . . . . 11 ⊢ ((H:A–1-1→B ∧ u ∈ A) → (∃y ∈ A ((H ‘u) = (H ‘y) ∧ vRy) ↔ ∃y ∈ A (y = u ∧ vRy)))
39		breq2 4644	. . . . . . . . . . . . 13 ⊢ (y = u → (vRy ↔ vRu))
40	39	ceqsrexv 2973	. . . . . . . . . . . 12 ⊢ (u ∈ A → (∃y ∈ A (y = u ∧ vRy) ↔ vRu))
41	40	adantl 452	. . . . . . . . . . 11 ⊢ ((H:A–1-1→B ∧ u ∈ A) → (∃y ∈ A (y = u ∧ vRy) ↔ vRu))
42	38, 41	bitrd 244	. . . . . . . . . 10 ⊢ ((H:A–1-1→B ∧ u ∈ A) → (∃y ∈ A ((H ‘u) = (H ‘y) ∧ vRy) ↔ vRu))
43	32, 42	sylan9bb 680	. . . . . . . . 9 ⊢ (((H:A–1-1→B ∧ v ∈ A) ∧ (H:A–1-1→B ∧ u ∈ A)) → (∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ vRu))
44	43	anandis 803	. . . . . . . 8 ⊢ ((H:A–1-1→B ∧ (v ∈ A ∧ u ∈ A)) → (∃x ∈ A ∃y ∈ A (((H ‘v) = (H ‘x) ∧ (H ‘u) = (H ‘y)) ∧ xRy) ↔ vRu))
45	15, 44	syl5bb 248	. . . . . . 7 ⊢ ((H:A–1-1→B ∧ (v ∈ A ∧ u ∈ A)) → (⟨(H ‘v), (H ‘u)⟩ ∈ {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)} ↔ vRu))
46	4, 45	sylan9bbr 681	. . . . . 6 ⊢ (((H:A–1-1→B ∧ (v ∈ A ∧ u ∈ A)) ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) → (⟨(H ‘v), (H ‘u)⟩ ∈ S ↔ vRu))
47	46	an32s 779	. . . . 5 ⊢ (((H:A–1-1→B ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) ∧ (v ∈ A ∧ u ∈ A)) → (⟨(H ‘v), (H ‘u)⟩ ∈ S ↔ vRu))
48	3, 47	syl5rbb 249	. . . 4 ⊢ (((H:A–1-1→B ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) ∧ (v ∈ A ∧ u ∈ A)) → (vRu ↔ (H ‘v)S(H ‘u)))
49	48	ralrimivva 2707	. . 3 ⊢ ((H:A–1-1→B ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) → ∀v ∈ A ∀u ∈ A (vRu ↔ (H ‘v)S(H ‘u)))
50	2, 49	sylan 457	. 2 ⊢ ((H:A–1-1-onto→B ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) → ∀v ∈ A ∀u ∈ A (vRu ↔ (H ‘v)S(H ‘u)))
51		df-iso 4797	. 2 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀v ∈ A ∀u ∈ A (vRu ↔ (H ‘v)S(H ‘u))))
52	1, 50, 51	sylanbrc 645	1 ⊢ ((H:A–1-1-onto→B ∧ S = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (H ‘x) ∧ w = (H ‘y)) ∧ xRy)}) → H Isom R, S (A, B))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  ⟨cop 4562  {copab 4623   class class class wbr 4640  –1-1→wf1 4779  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

This proof depends on 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 proof 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-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is used by:  f1oiso2  5501