Theorem f1oiso 5408

Description: Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 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 102	. 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 5068	. . 3 ⊢ (𝐻:A–1-1-onto→B → 𝐻:A–1-1→B)
3		df-br 3756	. . . . 5 ⊢ ((𝐻‘v)𝑆(𝐻‘u) ↔ ⟨(𝐻‘v), (𝐻‘u)⟩ ∈ 𝑆)
4		eleq2 2098	. . . . . . 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		f1fn 5036	. . . . . . . . 9 ⊢ (𝐻:A–1-1→B → 𝐻 Fn A)
6		funfvex 5135	. . . . . . . . . . . 12 ⊢ ((Fun 𝐻 ∧ v ∈ dom 𝐻) → (𝐻‘v) ∈ V)
7	6	funfni 4942	. . . . . . . . . . 11 ⊢ ((𝐻 Fn A ∧ v ∈ A) → (𝐻‘v) ∈ V)
8		funfvex 5135	. . . . . . . . . . . 12 ⊢ ((Fun 𝐻 ∧ u ∈ dom 𝐻) → (𝐻‘u) ∈ V)
9	8	funfni 4942	. . . . . . . . . . 11 ⊢ ((𝐻 Fn A ∧ u ∈ A) → (𝐻‘u) ∈ V)
10	7, 9	anim12dan 532	. . . . . . . . . 10 ⊢ ((𝐻 Fn A ∧ (v ∈ A ∧ u ∈ A)) → ((𝐻‘v) ∈ V ∧ (𝐻‘u) ∈ V))
11		eqeq1 2043	. . . . . . . . . . . . . 14 ⊢ (z = (𝐻‘v) → (z = (𝐻‘x) ↔ (𝐻‘v) = (𝐻‘x)))
12	11	anbi1d 438	. . . . . . . . . . . . 13 ⊢ (z = (𝐻‘v) → ((z = (𝐻‘x) ∧ w = (𝐻‘y)) ↔ ((𝐻‘v) = (𝐻‘x) ∧ w = (𝐻‘y))))
13	12	anbi1d 438	. . . . . . . . . . . 12 ⊢ (z = (𝐻‘v) → (((z = (𝐻‘x) ∧ w = (𝐻‘y)) ∧ x𝑅y) ↔ (((𝐻‘v) = (𝐻‘x) ∧ w = (𝐻‘y)) ∧ x𝑅y)))
14	13	2rexbidv 2343	. . . . . . . . . . 11 ⊢ (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)))
15		eqeq1 2043	. . . . . . . . . . . . . 14 ⊢ (w = (𝐻‘u) → (w = (𝐻‘y) ↔ (𝐻‘u) = (𝐻‘y)))
16	15	anbi2d 437	. . . . . . . . . . . . 13 ⊢ (w = (𝐻‘u) → (((𝐻‘v) = (𝐻‘x) ∧ w = (𝐻‘y)) ↔ ((𝐻‘v) = (𝐻‘x) ∧ (𝐻‘u) = (𝐻‘y))))
17	16	anbi1d 438	. . . . . . . . . . . 12 ⊢ (w = (𝐻‘u) → ((((𝐻‘v) = (𝐻‘x) ∧ w = (𝐻‘y)) ∧ x𝑅y) ↔ (((𝐻‘v) = (𝐻‘x) ∧ (𝐻‘u) = (𝐻‘y)) ∧ x𝑅y)))
18	17	2rexbidv 2343	. . . . . . . . . . 11 ⊢ (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)))
19	14, 18	opelopabg 3996	. . . . . . . . . 10 ⊢ (((𝐻‘v) ∈ V ∧ (𝐻‘u) ∈ V) → (⟨(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)))
20	10, 19	syl 14	. . . . . . . . 9 ⊢ ((𝐻 Fn A ∧ (v ∈ A ∧ u ∈ A)) → (⟨(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)))
21	5, 20	sylan 267	. . . . . . . 8 ⊢ ((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)} ↔ ∃x ∈ A ∃y ∈ A (((𝐻‘v) = (𝐻‘x) ∧ (𝐻‘u) = (𝐻‘y)) ∧ x𝑅y)))
22		anass 381	. . . . . . . . . . . . . . 15 ⊢ ((((𝐻‘v) = (𝐻‘x) ∧ (𝐻‘u) = (𝐻‘y)) ∧ x𝑅y) ↔ ((𝐻‘v) = (𝐻‘x) ∧ ((𝐻‘u) = (𝐻‘y) ∧ x𝑅y)))
23		f1fveq 5354	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻:A–1-1→B ∧ (v ∈ A ∧ x ∈ A)) → ((𝐻‘v) = (𝐻‘x) ↔ v = x))
24		equcom 1590	. . . . . . . . . . . . . . . . . 18 ⊢ (v = x ↔ x = v)
25	23, 24	syl6bb 185	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻:A–1-1→B ∧ (v ∈ A ∧ x ∈ A)) → ((𝐻‘v) = (𝐻‘x) ↔ x = v))
26	25	anassrs 380	. . . . . . . . . . . . . . . 16 ⊢ (((𝐻:A–1-1→B ∧ v ∈ A) ∧ x ∈ A) → ((𝐻‘v) = (𝐻‘x) ↔ x = v))
27	26	anbi1d 438	. . . . . . . . . . . . . . 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))))
28	22, 27	syl5bb 181	. . . . . . . . . . . . . 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))))
29	28	rexbidv 2321	. . . . . . . . . . . . 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))))
30		r19.42v 2461	. . . . . . . . . . . . 13 ⊢ (∃y ∈ A (x = v ∧ ((𝐻‘u) = (𝐻‘y) ∧ x𝑅y)) ↔ (x = v ∧ ∃y ∈ A ((𝐻‘u) = (𝐻‘y) ∧ x𝑅y)))
31	29, 30	syl6bb 185	. . . . . . . . . . . 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))))
32	31	rexbidva 2317	. . . . . . . . . . 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))))
33		breq1 3758	. . . . . . . . . . . . . . 15 ⊢ (x = v → (x𝑅y ↔ v𝑅y))
34	33	anbi2d 437	. . . . . . . . . . . . . 14 ⊢ (x = v → (((𝐻‘u) = (𝐻‘y) ∧ x𝑅y) ↔ ((𝐻‘u) = (𝐻‘y) ∧ v𝑅y)))
35	34	rexbidv 2321	. . . . . . . . . . . . 13 ⊢ (x = v → (∃y ∈ A ((𝐻‘u) = (𝐻‘y) ∧ x𝑅y) ↔ ∃y ∈ A ((𝐻‘u) = (𝐻‘y) ∧ v𝑅y)))
36	35	ceqsrexv 2668	. . . . . . . . . . . 12 ⊢ (v ∈ A → (∃x ∈ A (x = v ∧ ∃y ∈ A ((𝐻‘u) = (𝐻‘y) ∧ x𝑅y)) ↔ ∃y ∈ A ((𝐻‘u) = (𝐻‘y) ∧ v𝑅y)))
37	36	adantl 262	. . . . . . . . . . 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)))
38	32, 37	bitrd 177	. . . . . . . . . 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)))
39		f1fveq 5354	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:A–1-1→B ∧ (u ∈ A ∧ y ∈ A)) → ((𝐻‘u) = (𝐻‘y) ↔ u = y))
40		equcom 1590	. . . . . . . . . . . . . . 15 ⊢ (u = y ↔ y = u)
41	39, 40	syl6bb 185	. . . . . . . . . . . . . 14 ⊢ ((𝐻:A–1-1→B ∧ (u ∈ A ∧ y ∈ A)) → ((𝐻‘u) = (𝐻‘y) ↔ y = u))
42	41	anassrs 380	. . . . . . . . . . . . 13 ⊢ (((𝐻:A–1-1→B ∧ u ∈ A) ∧ y ∈ A) → ((𝐻‘u) = (𝐻‘y) ↔ y = u))
43	42	anbi1d 438	. . . . . . . . . . . 12 ⊢ (((𝐻:A–1-1→B ∧ u ∈ A) ∧ y ∈ A) → (((𝐻‘u) = (𝐻‘y) ∧ v𝑅y) ↔ (y = u ∧ v𝑅y)))
44	43	rexbidva 2317	. . . . . . . . . . 11 ⊢ ((𝐻:A–1-1→B ∧ u ∈ A) → (∃y ∈ A ((𝐻‘u) = (𝐻‘y) ∧ v𝑅y) ↔ ∃y ∈ A (y = u ∧ v𝑅y)))
45		breq2 3759	. . . . . . . . . . . . 13 ⊢ (y = u → (v𝑅y ↔ v𝑅u))
46	45	ceqsrexv 2668	. . . . . . . . . . . 12 ⊢ (u ∈ A → (∃y ∈ A (y = u ∧ v𝑅y) ↔ v𝑅u))
47	46	adantl 262	. . . . . . . . . . 11 ⊢ ((𝐻:A–1-1→B ∧ u ∈ A) → (∃y ∈ A (y = u ∧ v𝑅y) ↔ v𝑅u))
48	44, 47	bitrd 177	. . . . . . . . . 10 ⊢ ((𝐻:A–1-1→B ∧ u ∈ A) → (∃y ∈ A ((𝐻‘u) = (𝐻‘y) ∧ v𝑅y) ↔ v𝑅u))
49	38, 48	sylan9bb 435	. . . . . . . . 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))
50	49	anandis 526	. . . . . . . 8 ⊢ ((𝐻:A–1-1→B ∧ (v ∈ A ∧ u ∈ A)) → (∃x ∈ A ∃y ∈ A (((𝐻‘v) = (𝐻‘x) ∧ (𝐻‘u) = (𝐻‘y)) ∧ x𝑅y) ↔ v𝑅u))
51	21, 50	bitrd 177	. . . . . . 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))
52	4, 51	sylan9bbr 436	. . . . . 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))
53	52	an32s 502	. . . . 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))
54	3, 53	syl5rbb 182	. . . 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)))
55	54	ralrimivva 2395	. . 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)))
56	2, 55	sylan 267	. 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)))
57		df-isom 4854	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A–1-1-onto→B ∧ ∀v ∈ A ∀u ∈ A (v𝑅u ↔ (𝐻‘v)𝑆(𝐻‘u))))
58	1, 56, 57	sylanbrc 394	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))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  Vcvv 2551  ⟨cop 3370   class class class wbr 3755  {copab 3808   Fn wfn 4840  –1-1→wf1 4842  –1-1-onto→wf1o 4844  ‘cfv 4845   Isom wiso 4846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-f1o 4852  df-fv 4853  df-isom 4854

This theorem is referenced by:  f1oiso2  5409