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Theorem isose 7379

Description: An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)

**Assertion**
Ref	Expression
isose	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Proof of Theorem isose Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		isof1o 7359	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1ofun 6864	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻)
4		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
5	4	funimaex 6666	. . . 4 ⊢ (Fun 𝐻 → (𝐻 “ 𝑥) ∈ V)
6	2, 3, 5	3syl 18	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ∈ V)
7	1, 6	isoselem 7377	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
8		isocnv 7366	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
9		isof1o 7359	. . . 4 ⊢ (^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ^◡𝐻:𝐵–1-1-onto→𝐴)
10		f1ofun 6864	. . . 4 ⊢ (^◡𝐻:𝐵–1-1-onto→𝐴 → Fun ^◡𝐻)
11	4	funimaex 6666	. . . 4 ⊢ (Fun ^◡𝐻 → (^◡𝐻 “ 𝑥) ∈ V)
12	8, 9, 10, 11	4syl 19	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (^◡𝐻 “ 𝑥) ∈ V)
13	8, 12	isoselem 7377	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Se 𝐵 → 𝑅 Se 𝐴))
14	7, 13	impbid 212	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Vcvv 3488 Se wse 5650 ^◡ccnv 5699 “ cima 5703 Fun wfun 6567 –1-1-onto→wf1o 6572 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-rep 5303 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-nfc 2895 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-mpt 5250 df-id 5593 df-se 5653 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: (None)