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Mirrors > Home > MPE Home > Th. List > isose | Structured version Visualization version GIF version |
Description: An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Ref | Expression |
---|---|
isose | ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | |
2 | isof1o 7327 | . . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵) | |
3 | f1ofun 6837 | . . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻) | |
4 | vex 3466 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 4 | funimaex 6639 | . . . 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 7345 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵)) |
8 | isocnv 7334 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | |
9 | isof1o 7327 | . . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻:𝐵–1-1-onto→𝐴) | |
10 | f1ofun 6837 | . . . 4 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → Fun ◡𝐻) | |
11 | 4 | funimaex 6639 | . . . 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 7345 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Se 𝐵 → 𝑅 Se 𝐴)) |
14 | 7, 13 | impbid 211 | 1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 Vcvv 3462 Se wse 5627 ◡ccnv 5673 “ cima 5677 Fun wfun 6540 –1-1-onto→wf1o 6545 Isom wiso 6547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-se 5630 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 |
This theorem is referenced by: (None) |
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