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Theorem isose 7342

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 23	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		isof1o 7322	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1ofun 6823	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻)
4		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
5	4	funimaex 6624	. . . 4 ⊢ (Fun 𝐻 → (𝐻 “ 𝑥) ∈ V)
6	2, 3, 5	3syl 19	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ∈ V)
7	1, 6	isoselem 7340	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
8		isocnv 7329	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
9		isof1o 7322	. . . 4 ⊢ (^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ^◡𝐻:𝐵–1-1-onto→𝐴)
10		f1ofun 6823	. . . 4 ⊢ (^◡𝐻:𝐵–1-1-onto→𝐴 → Fun ^◡𝐻)
11	4	funimaex 6624	. . . 4 ⊢ (Fun ^◡𝐻 → (^◡𝐻 “ 𝑥) ∈ V)
12	8, 9, 10, 11	4syl 20	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (^◡𝐻 “ 𝑥) ∈ V)
13	8, 12	isoselem 7340	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Se 𝐵 → 𝑅 Se 𝐴))
14	7, 13	impbid 215	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2149 Vcvv 3463 Se wse 5613 ^◡ccnv 5661 “ cima 5665 Fun wfun 6531 –1-1-onto→wf1o 6536 Isom wiso 6538

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pr 5405

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-se 5616 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546

This theorem is referenced by: onsse 28432 vonf1osev 35529