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Theorem isose 7288

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 7268	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1ofun 6770	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻)
4		vex 3435	. . . . 5 ⊢ 𝑥 ∈ V
5	4	funimaex 6574	. . . 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 7286	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵))
8		isocnv 7275	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
9		isof1o 7268	. . . 4 ⊢ (^◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ^◡𝐻:𝐵–1-1-onto→𝐴)
10		f1ofun 6770	. . . 4 ⊢ (^◡𝐻:𝐵–1-1-onto→𝐴 → Fun ^◡𝐻)
11	4	funimaex 6574	. . . 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 7286	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Se 𝐵 → 𝑅 Se 𝐴))
14	7, 13	impbid 213	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2119 Vcvv 3431 Se wse 5570 ^◡ccnv 5618 “ cima 5622 Fun wfun 6480 –1-1-onto→wf1o 6485 Isom wiso 6487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pr 5363

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4263 df-if 4456 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5155 df-id 5514 df-se 5573 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495

This theorem is referenced by: onsse 28284