Theorem f1oiso2 7388

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 7321	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (◡𝐻‘𝑥) ∈ 𝐴)
3	2	adantrr 716	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐻‘𝑥) ∈ 𝐴)
4	3	3adant3 1132	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑥) ∈ 𝐴)
5		f1ocnvdm 7321	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (◡𝐻‘𝑦) ∈ 𝐴)
6	5	adantrl 715	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐻‘𝑦) ∈ 𝐴)
7	6	3adant3 1132	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑦) ∈ 𝐴)
8		f1ocnvfv2 7313	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑥)) = 𝑥)
9	8	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 = (𝐻‘(◡𝐻‘𝑥)))
10		f1ocnvfv2 7313	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑦)) = 𝑦)
11	10	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐻‘(◡𝐻‘𝑦)))
12	9, 11	anim12dan 618	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦))))
13	12	3adant3 1132	. . . . . . . 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 1138	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)) → (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))
15		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑤 = (◡𝐻‘𝑦) → (𝐻‘𝑤) = (𝐻‘(◡𝐻‘𝑦)))
16	15	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑤 = (◡𝐻‘𝑦) → (𝑦 = (𝐻‘𝑤) ↔ 𝑦 = (𝐻‘(◡𝐻‘𝑦))))
17	16	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐻‘𝑦) → ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ↔ (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘(◡𝐻‘𝑦)))))
18		breq2 5170	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐻‘𝑦) → ((◡𝐻‘𝑥)𝑅𝑤 ↔ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦)))
19	17, 18	anbi12d 631	. . . . . . . . 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 3635	. . . . . . . 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 836	. . . . . . 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 6920	. . . . . . . . . . . 12 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝐻‘𝑧) = (𝐻‘(◡𝐻‘𝑥)))
23	22	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝑥 = (𝐻‘𝑧) ↔ 𝑥 = (𝐻‘(◡𝐻‘𝑥))))
24	23	anbi1d 630	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐻‘𝑥) → ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ↔ (𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤))))
25		breq1 5169	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐻‘𝑥) → (𝑧𝑅𝑤 ↔ (◡𝐻‘𝑥)𝑅𝑤))
26	24, 25	anbi12d 631	. . . . . . . . 9 ⊢ (𝑧 = (◡𝐻‘𝑥) → (((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤) ↔ ((𝑥 = (𝐻‘(◡𝐻‘𝑥)) ∧ 𝑦 = (𝐻‘𝑤)) ∧ (◡𝐻‘𝑥)𝑅𝑤)))
27	26	rexbidv 3185	. . . . . . . 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 3635	. . . . . . 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 583	. . . . . 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 1122	. . . . 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 1244	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 = (𝐻‘𝑧))
32		simp1 1136	. . . . . . . . . 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 1199	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧 ∈ 𝐴)
34		f1of 6862	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
35	34	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐻‘𝑧) ∈ 𝐵)
36	32, 33, 35	syl2anc 583	. . . . . . . . 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 2844	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 ∈ 𝐵)
38		simp3lr 1245	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 = (𝐻‘𝑤))
39		simp2r 1200	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑤 ∈ 𝐴)
40	34	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝐵)
41	32, 39, 40	syl2anc 583	. . . . . . . . 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 2844	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 ∈ 𝐵)
43		simp3r 1202	. . . . . . . . 9 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝑅𝑤)
44	31	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑧) = 𝑥)
45		f1ocnvfv 7314	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) = 𝑥 → (◡𝐻‘𝑥) = 𝑧))
46	32, 33, 45	syl2anc 583	. . . . . . . . . 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 2746	. . . . . . . . . 10 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻‘𝑤) = 𝑦)
49		f1ocnvfv 7314	. . . . . . . . . . 11 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤) = 𝑦 → (◡𝐻‘𝑦) = 𝑤))
50	32, 39, 49	syl2anc 583	. . . . . . . . . 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 5198	. . . . . . . 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 514	. . . . . . 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 1119	. . . . . 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 3218	. . . . 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 212	. . . 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 5232	. . 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	eqtrid 2792	. 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝑥 = (𝐻‘𝑧) ∧ 𝑦 = (𝐻‘𝑤)) ∧ 𝑧𝑅𝑤)})
59		f1oiso 7387	. 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 686	1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166  {copab 5228  ^◡ccnv 5699  –1-1-onto→wf1o 6572  ‘cfv 6573   Isom wiso 6574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582

This theorem is referenced by:  fnwelem  8172