Theorem f1oiso2 6502

Description: Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)

Hypothesis

Ref	Expression
f1oiso2.1	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))}

Assertion

Ref	Expression
f1oiso2	⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))

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

Allowed substitution hints:   𝑆(𝑥,𝑦)

Proof of Theorem f1oiso2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oiso2.1	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))}
2		f1ocnvdm 6440	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (◡𝐻‘𝑥) ∈ 𝐴)
3	2	adantrr 749	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐻‘𝑥) ∈ 𝐴)
4	3	3adant3 1074	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑥) ∈ 𝐴)
5		f1ocnvdm 6440	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (◡𝐻‘𝑦) ∈ 𝐴)
6	5	adantrl 748	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐻‘𝑦) ∈ 𝐴)
7	6	3adant3 1074	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑦) ∈ 𝐴)
8		f1ocnvfv2 6433	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑥)) = 𝑥)
9	8	eqcomd 2616	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 = (𝐻‘(◡𝐻‘𝑥)))
10		f1ocnvfv2 6433	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑦)) = 𝑦)
11	10	eqcomd 2616	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐻‘(◡𝐻‘𝑦)))
12	9, 11	anim12dan 878	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦))))
13	12	3adant3 1074	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦))))
14		simp3 1056	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))
15		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑤 = (◡𝐻‘𝑦) → (𝐻‘𝑤) = (𝐻‘(◡𝐻‘𝑦)))
16	15	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑤 = (◡𝐻‘𝑦) → (𝑦 = (𝐻‘𝑤) ↔ 𝑦 = (𝐻‘(◡𝐻‘𝑦))))
17	16	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐻‘𝑦) → ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ↔ (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦)))))
18		breq2 4587	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐻‘𝑦) → ((◡𝐻‘𝑥)𝑅𝑤 ↔ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)))
19	17, 18	anbi12d 743	. . . . . . . . 9 ⊢ (𝑤 = (◡𝐻‘𝑦) → (((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤) ↔ ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦))) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))))
20	19	rspcev 3282	. . . . . . . 8 ⊢ (((◡𝐻‘𝑦) ∈ 𝐴 ∧ ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦))) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))) → ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤))
21	7, 13, 14, 20	syl12anc 1316	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤))
22		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝐻‘𝑧) = (𝐻‘(◡𝐻‘𝑥)))
23	22	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝑥 = (𝐻‘𝑧) ↔ 𝑥 = (𝐻‘(◡𝐻‘𝑥))))
24	23	anbi1d 737	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐻‘𝑥) → ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ↔ (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤))))
25		breq1 4586	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝑧𝑅𝑤 ↔ (◡𝐻‘𝑥)𝑅𝑤))
26	24, 25	anbi12d 743	. . . . . . . . 9 ⊢ (𝑧 = (◡𝐻‘𝑥) → (((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤) ↔ ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤)))
27	26	rexbidv 3034	. . . . . . . 8 ⊢ (𝑧 = (◡𝐻‘𝑥) → (∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤) ↔ ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤)))
28	27	rspcev 3282	. . . . . . 7 ⊢ (((◡𝐻‘𝑥) ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤)) → ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤))
29	4, 21, 28	syl2anc 691	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤))
30	29	3expib 1260	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)))
31		simp3ll 1125	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 = (𝐻‘𝑧))
32		simp1 1054	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝐻:𝐴–1-1-onto→𝐵)
33		simp2l 1080	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧 ∈ 𝐴)
34		f1of 6050	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
35	34	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐻‘𝑧) ∈ 𝐵)
36	32, 33, 35	syl2anc 691	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑧) ∈ 𝐵)
37	31, 36	eqeltrd 2688	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 ∈ 𝐵)
38		simp3lr 1126	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 = (𝐻‘𝑤))
39		simp2r 1081	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑤 ∈ 𝐴)
40	34	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝐵)
41	32, 39, 40	syl2anc 691	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑤) ∈ 𝐵)
42	38, 41	eqeltrd 2688	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 ∈ 𝐵)
43		simp3r 1083	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝑅𝑤)
44	31	eqcomd 2616	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑧) = 𝑥)
45		f1ocnvfv 6434	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) = 𝑥 → (◡𝐻‘𝑥) = 𝑧))
46	32, 33, 45	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝐻‘𝑧) = 𝑥 → (◡𝐻‘𝑥) = 𝑧))
47	44, 46	mpd 15	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (◡𝐻‘𝑥) = 𝑧)
48	38	eqcomd 2616	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑤) = 𝑦)
49		f1ocnvfv 6434	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤) = 𝑦 → (◡𝐻‘𝑦) = 𝑤))
50	32, 39, 49	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝐻‘𝑤) = 𝑦 → (◡𝐻‘𝑦) = 𝑤))
51	48, 50	mpd 15	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (◡𝐻‘𝑦) = 𝑤)
52	43, 47, 51	3brtr4d 4615	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))
53	37, 42, 52	jca31 555	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)))
54	53	3exp 1256	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)))))
55	54	rexlimdvv 3019	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))))
56	30, 55	impbid 201	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)))
57	56	opabbidv 4648	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)})
58	1, 57	syl5eq 2656	. 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)})
59		f1oiso 6501	. 2 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
60	58, 59	mpdan 699	1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583  {copab 4642  ^◡ccnv 5037  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813

This theorem is referenced by:  fnwelem  7179