Theorem f1oiso2 5567

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 5521	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (◡𝐻‘𝑥) ∈ 𝐴)
3	2	adantrr 463	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐻‘𝑥) ∈ 𝐴)
4	3	3adant3 961	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑥) ∈ 𝐴)
5		f1ocnvdm 5521	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (◡𝐻‘𝑦) ∈ 𝐴)
6	5	adantrl 462	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐻‘𝑦) ∈ 𝐴)
7	6	3adant3 961	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑦) ∈ 𝐴)
8		f1ocnvfv2 5518	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑥)) = 𝑥)
9	8	eqcomd 2090	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 = (𝐻‘(◡𝐻‘𝑥)))
10		f1ocnvfv2 5518	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑦)) = 𝑦)
11	10	eqcomd 2090	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐻‘(◡𝐻‘𝑦)))
12	9, 11	anim12dan 565	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦))))
13	12	3adant3 961	. . . . . . . 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 943	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))
15		fveq2 5268	. . . . . . . . . . . 12 ⊢ (𝑤 = (◡𝐻‘𝑦) → (𝐻‘𝑤) = (𝐻‘(◡𝐻‘𝑦)))
16	15	eqeq2d 2096	. . . . . . . . . . 11 ⊢ (𝑤 = (◡𝐻‘𝑦) → (𝑦 = (𝐻‘𝑤) ↔ 𝑦 = (𝐻‘(◡𝐻‘𝑦))))
17	16	anbi2d 452	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐻‘𝑦) → ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ↔ (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦)))))
18		breq2 3824	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐻‘𝑦) → ((◡𝐻‘𝑥)𝑅𝑤 ↔ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)))
19	17, 18	anbi12d 457	. . . . . . . . 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 2715	. . . . . . . 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 1170	. . . . . . 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 5268	. . . . . . . . . . . 12 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝐻‘𝑧) = (𝐻‘(◡𝐻‘𝑥)))
23	22	eqeq2d 2096	. . . . . . . . . . 11 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝑥 = (𝐻‘𝑧) ↔ 𝑥 = (𝐻‘(◡𝐻‘𝑥))))
24	23	anbi1d 453	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐻‘𝑥) → ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ↔ (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤))))
25		breq1 3823	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝑧𝑅𝑤 ↔ (◡𝐻‘𝑥)𝑅𝑤))
26	24, 25	anbi12d 457	. . . . . . . . 9 ⊢ (𝑧 = (◡𝐻‘𝑥) → (((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤) ↔ ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤)))
27	26	rexbidv 2377	. . . . . . . 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 2715	. . . . . . 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 403	. . . . . 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 1144	. . . . 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 1012	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 = (𝐻‘𝑧))
32		simp1 941	. . . . . . . . . 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 967	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧 ∈ 𝐴)
34		f1of 5216	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
35	34	ffvelrnda 5397	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐻‘𝑧) ∈ 𝐵)
36	32, 33, 35	syl2anc 403	. . . . . . . . 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 2161	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 ∈ 𝐵)
38		simp3lr 1013	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 = (𝐻‘𝑤))
39		simp2r 968	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑤 ∈ 𝐴)
40	34	ffvelrnda 5397	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝐵)
41	32, 39, 40	syl2anc 403	. . . . . . . . 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 2161	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 ∈ 𝐵)
43		simp3r 970	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝑅𝑤)
44	31	eqcomd 2090	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑧) = 𝑥)
45		f1ocnvfv 5519	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) = 𝑥 → (◡𝐻‘𝑥) = 𝑧))
46	32, 33, 45	syl2anc 403	. . . . . . . . . 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 13	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (◡𝐻‘𝑥) = 𝑧)
48	38	eqcomd 2090	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑤) = 𝑦)
49		f1ocnvfv 5519	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤) = 𝑦 → (◡𝐻‘𝑦) = 𝑤))
50	32, 39, 49	syl2anc 403	. . . . . . . . . 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 13	. . . . . . . . 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 3850	. . . . . . . 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 302	. . . . . . 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 1140	. . . . . 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 2491	. . . . 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 127	. . . 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 3879	. . 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 2129	. 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)})
59		f1oiso 5566	. 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 412	1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 922   = wceq 1287   ∈ wcel 1436  ∃wrex 2356   class class class wbr 3820  {copab 3873  ^◡ccnv 4410  –1-1-onto→wf1o 4980  ‘cfv 4981   Isom wiso 4982

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-isom 4990

This theorem is referenced by: (None)