Theorem f1oiso 5485

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 107	. 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 5145	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
3		df-br 3786	. . . . 5 ⊢ ((𝐻‘𝑣)𝑆(𝐻‘𝑢) ↔ ⟨(𝐻‘𝑣), (𝐻‘𝑢)⟩ ∈ 𝑆)
4		eleq2 2142	. . . . . . 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 5113	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1→𝐵 → 𝐻 Fn 𝐴)
6		funfvex 5212	. . . . . . . . . . . 12 ⊢ ((Fun 𝐻 ∧ 𝑣 ∈ dom 𝐻) → (𝐻‘𝑣) ∈ V)
7	6	funfni 5019	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) ∈ V)
8		funfvex 5212	. . . . . . . . . . . 12 ⊢ ((Fun 𝐻 ∧ 𝑢 ∈ dom 𝐻) → (𝐻‘𝑢) ∈ V)
9	8	funfni 5019	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝐻‘𝑢) ∈ V)
10	7, 9	anim12dan 564	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → ((𝐻‘𝑣) ∈ V ∧ (𝐻‘𝑢) ∈ V))
11		eqeq1 2087	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻‘𝑣) → (𝑧 = (𝐻‘𝑥) ↔ (𝐻‘𝑣) = (𝐻‘𝑥)))
12	11	anbi1d 452	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐻‘𝑣) → ((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦))))
13	12	anbi1d 452	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑣) → (((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)))
14	13	2rexbidv 2391	. . . . . . . . . . 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 2087	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐻‘𝑢) → (𝑤 = (𝐻‘𝑦) ↔ (𝐻‘𝑢) = (𝐻‘𝑦)))
16	15	anbi2d 451	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐻‘𝑢) → (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦))))
17	16	anbi1d 452	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑢) → ((((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)))
18	17	2rexbidv 2391	. . . . . . . . . . 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 4023	. . . . . . . . . 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 277	. . . . . . . 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 393	. . . . . . . . . . . . . . 15 ⊢ ((((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)))
23		f1fveq 5432	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑣 = 𝑥))
24		equcom 1633	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑥 ↔ 𝑥 = 𝑣)
25	23, 24	syl6bb 194	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑥 = 𝑣))
26	25	anassrs 392	. . . . . . . . . . . . . . . 16 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑥 = 𝑣))
27	26	anbi1d 452	. . . . . . . . . . . . . . 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 190	. . . . . . . . . . . . . 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 2369	. . . . . . . . . . . . 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 2511	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 = 𝑣 ∧ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑣 ∧ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)))
31	29, 30	syl6bb 194	. . . . . . . . . . . 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 2365	. . . . . . . . . . 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 3788	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → (𝑥𝑅𝑦 ↔ 𝑣𝑅𝑦))
34	33	anbi2d 451	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑣 → (((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦) ↔ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
35	34	rexbidv 2369	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
36	35	ceqsrexv 2725	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝑥 = 𝑣 ∧ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
37	36	adantl 271	. . . . . . . . . . 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 186	. . . . . . . . . 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 5432	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑢 = 𝑦))
40		equcom 1633	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 ↔ 𝑦 = 𝑢)
41	39, 40	syl6bb 194	. . . . . . . . . . . . . 14 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑦 = 𝑢))
42	41	anassrs 392	. . . . . . . . . . . . 13 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑦 = 𝑢))
43	42	anbi1d 452	. . . . . . . . . . . 12 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦)))
44	43	rexbidva 2365	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦)))
45		breq2 3789	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑢 → (𝑣𝑅𝑦 ↔ 𝑣𝑅𝑢))
46	45	ceqsrexv 2725	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝐴 → (∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
47	46	adantl 271	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
48	44, 47	bitrd 186	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
49	38, 48	sylan9bb 449	. . . . . . . . 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 556	. . . . . . . 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 186	. . . . . . 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 450	. . . . . 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 532	. . . . 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 191	. . . 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 2443	. . 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 277	. 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 4931	. 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 408	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 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  Vcvv 2601  ⟨cop 3401   class class class wbr 3785  {copab 3838   Fn wfn 4917  –1-1→wf1 4919  –1-1-onto→wf1o 4921  ‘cfv 4922   Isom wiso 4923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-f1o 4929  df-fv 4930  df-isom 4931

This theorem is referenced by:  f1oiso2  5486