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Theorem isose 6819

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 6799	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1ofun 6356	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻)
4		vex 3386	. . . . 5 ⊢ 𝑥 ∈ V
5	4	funimaex 6185	. . . 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 6817	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
8		isocnv 6806	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
9		isof1o 6799	. . . 4 ⊢ (^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ^◡𝐻:𝐵–1-1-onto→𝐴)
10		f1ofun 6356	. . . 4 ⊢ (^◡𝐻:𝐵–1-1-onto→𝐴 → Fun ^◡𝐻)
11	4	funimaex 6185	. . . 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 6817	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Se 𝐵 → 𝑅 Se 𝐴))
14	7, 13	impbid 204	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2157 Vcvv 3383 Se wse 5267 ^◡ccnv 5309 “ cima 5313 Fun wfun 6093 –1-1-onto→wf1o 6098 Isom wiso 6100

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-se 5270 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108

This theorem is referenced by: (None)