Theorem f1oiso 7222

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 483	. 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 6715	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
3		df-br 5075	. . . . 5 ⊢ ((𝐻‘𝑣)𝑆(𝐻‘𝑢) ↔ ⟨(𝐻‘𝑣), (𝐻‘𝑢)⟩ ∈ 𝑆)
4		eleq2 2827	. . . . . . 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 6787	. . . . . . . . 9 ⊢ (𝐻‘𝑣) ∈ V
6		fvex 6787	. . . . . . . . 9 ⊢ (𝐻‘𝑢) ∈ V
7		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑣) → (𝑧 = (𝐻‘𝑥) ↔ (𝐻‘𝑣) = (𝐻‘𝑥)))
8	7	anbi1d 630	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻‘𝑣) → ((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦))))
9	8	anbi1d 630	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻‘𝑣) → (((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)))
10	9	2rexbidv 3229	. . . . . . . . 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 2742	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑢) → (𝑤 = (𝐻‘𝑦) ↔ (𝐻‘𝑢) = (𝐻‘𝑦)))
12	11	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐻‘𝑢) → (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦))))
13	12	anbi1d 630	. . . . . . . . . 10 ⊢ (𝑤 = (𝐻‘𝑢) → ((((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)))
14	13	2rexbidv 3229	. . . . . . . . 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 5455	. . . . . . . 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 469	. . . . . . . . . . . . . . 15 ⊢ ((((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)))
17		f1fveq 7135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑣 = 𝑥))
18		equcom 2021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑥 ↔ 𝑥 = 𝑣)
19	17, 18	bitrdi 287	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑥 = 𝑣))
20	19	anassrs 468	. . . . . . . . . . . . . . . 16 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑥 = 𝑣))
21	20	anbi1d 630	. . . . . . . . . . . . . . 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	bitrid 282	. . . . . . . . . . . . . 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 3226	. . . . . . . . . . . . 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 3279	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 = 𝑣 ∧ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑣 ∧ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)))
25	23, 24	bitrdi 287	. . . . . . . . . . . 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 3225	. . . . . . . . . . 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 5077	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → (𝑥𝑅𝑦 ↔ 𝑣𝑅𝑦))
28	27	anbi2d 629	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑣 → (((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦) ↔ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
29	28	rexbidv 3226	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
30	29	ceqsrexv 3585	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝑥 = 𝑣 ∧ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
31	30	adantl 482	. . . . . . . . . . 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 278	. . . . . . . . . 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 7135	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑢 = 𝑦))
34		equcom 2021	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 ↔ 𝑦 = 𝑢)
35	33, 34	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑦 = 𝑢))
36	35	anassrs 468	. . . . . . . . . . . . 13 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑦 = 𝑢))
37	36	anbi1d 630	. . . . . . . . . . . 12 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦)))
38	37	rexbidva 3225	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦)))
39		breq2 5078	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑢 → (𝑣𝑅𝑦 ↔ 𝑣𝑅𝑢))
40	39	ceqsrexv 3585	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝐴 → (∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
41	40	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
42	38, 41	bitrd 278	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
43	32, 42	sylan9bb 510	. . . . . . . . 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 675	. . . . . . . 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	bitrid 282	. . . . . . 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 511	. . . . . 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 649	. . . . 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	bitr2id 284	. . . 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 3123	. . 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 580	. 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-isom 6442	. 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 583	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   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ⟨cop 4567   class class class wbr 5074  {copab 5136  –1-1→wf1 6430  –1-1-onto→wf1o 6432  ‘cfv 6433   Isom wiso 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-f1o 6440  df-fv 6441  df-isom 6442

This theorem is referenced by:  f1oiso2  7223  hartogslem1  9301  cnso  15956