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Theorem isose 7318

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 7298	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1ofun 6802	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻)
4		vex 3451	. . . . 5 ⊢ 𝑥 ∈ V
5	4	funimaex 6605	. . . 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 7316	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
8		isocnv 7305	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
9		isof1o 7298	. . . 4 ⊢ (^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ^◡𝐻:𝐵–1-1-onto→𝐴)
10		f1ofun 6802	. . . 4 ⊢ (^◡𝐻:𝐵–1-1-onto→𝐴 → Fun ^◡𝐻)
11	4	funimaex 6605	. . . 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 7316	. 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 2109 Vcvv 3447 Se wse 5589 ^◡ccnv 5637 “ cima 5641 Fun wfun 6505 –1-1-onto→wf1o 6510 Isom wiso 6512

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-se 5592 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520

This theorem is referenced by: onsse 28171