Theorem f1oiso 5873

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 109	. 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 5503	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
3		df-br 4034	. . . . 5 ⊢ ((𝐻‘𝑣)𝑆(𝐻‘𝑢) ↔ ⟨(𝐻‘𝑣), (𝐻‘𝑢)⟩ ∈ 𝑆)
4		eleq2 2260	. . . . . . 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 5465	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1→𝐵 → 𝐻 Fn 𝐴)
6		funfvex 5575	. . . . . . . . . . . 12 ⊢ ((Fun 𝐻 ∧ 𝑣 ∈ dom 𝐻) → (𝐻‘𝑣) ∈ V)
7	6	funfni 5358	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) ∈ V)
8		funfvex 5575	. . . . . . . . . . . 12 ⊢ ((Fun 𝐻 ∧ 𝑢 ∈ dom 𝐻) → (𝐻‘𝑢) ∈ V)
9	8	funfni 5358	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝐻‘𝑢) ∈ V)
10	7, 9	anim12dan 600	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → ((𝐻‘𝑣) ∈ V ∧ (𝐻‘𝑢) ∈ V))
11		eqeq1 2203	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻‘𝑣) → (𝑧 = (𝐻‘𝑥) ↔ (𝐻‘𝑣) = (𝐻‘𝑥)))
12	11	anbi1d 465	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐻‘𝑣) → ((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦))))
13	12	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑣) → (((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)))
14	13	2rexbidv 2522	. . . . . . . . . . 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 2203	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐻‘𝑢) → (𝑤 = (𝐻‘𝑦) ↔ (𝐻‘𝑢) = (𝐻‘𝑦)))
16	15	anbi2d 464	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐻‘𝑢) → (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦))))
17	16	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐻‘𝑢) → ((((𝐻‘𝑣) = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)))
18	17	2rexbidv 2522	. . . . . . . . . . 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 4302	. . . . . . . . . 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 283	. . . . . . . 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 401	. . . . . . . . . . . . . . 15 ⊢ ((((𝐻‘𝑣) = (𝐻‘𝑥) ∧ (𝐻‘𝑢) = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦) ↔ ((𝐻‘𝑣) = (𝐻‘𝑥) ∧ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)))
23		f1fveq 5819	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑣 = 𝑥))
24		equcom 1720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑥 ↔ 𝑥 = 𝑣)
25	23, 24	bitrdi 196	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑥 = 𝑣))
26	25	anassrs 400	. . . . . . . . . . . . . . . 16 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐻‘𝑣) = (𝐻‘𝑥) ↔ 𝑥 = 𝑣))
27	26	anbi1d 465	. . . . . . . . . . . . . . 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	bitrid 192	. . . . . . . . . . . . . 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 2498	. . . . . . . . . . . . 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 2654	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 = 𝑣 ∧ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑣 ∧ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)))
31	29, 30	bitrdi 196	. . . . . . . . . . . 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 2494	. . . . . . . . . . 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 4036	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → (𝑥𝑅𝑦 ↔ 𝑣𝑅𝑦))
34	33	anbi2d 464	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑣 → (((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦) ↔ ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
35	34	rexbidv 2498	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
36	35	ceqsrexv 2894	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝑥 = 𝑣 ∧ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦)))
37	36	adantl 277	. . . . . . . . . . 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 188	. . . . . . . . . 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 5819	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑢 = 𝑦))
40		equcom 1720	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑦 ↔ 𝑦 = 𝑢)
41	39, 40	bitrdi 196	. . . . . . . . . . . . . 14 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑦 = 𝑢))
42	41	anassrs 400	. . . . . . . . . . . . 13 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑢) = (𝐻‘𝑦) ↔ 𝑦 = 𝑢))
43	42	anbi1d 465	. . . . . . . . . . . 12 ⊢ (((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦)))
44	43	rexbidva 2494	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦)))
45		breq2 4037	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑢 → (𝑣𝑅𝑦 ↔ 𝑣𝑅𝑢))
46	45	ceqsrexv 2894	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝐴 → (∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
47	46	adantl 277	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 (𝑦 = 𝑢 ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
48	44, 47	bitrd 188	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝑢 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 ((𝐻‘𝑢) = (𝐻‘𝑦) ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
49	38, 48	sylan9bb 462	. . . . . . . . 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 592	. . . . . . . 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 188	. . . . . . 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 463	. . . . . 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 568	. . . . 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	bitr2id 193	. . . 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 2579	. . 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 283	. 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 5267	. 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 417	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 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  Vcvv 2763  ⟨cop 3625   class class class wbr 4033  {copab 4093   Fn wfn 5253  –1-1→wf1 5255  –1-1-onto→wf1o 5257  ‘cfv 5258   Isom wiso 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-f1o 5265  df-fv 5266  df-isom 5267

This theorem is referenced by:  f1oiso2  5874