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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 6473 | . . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵) | |
3 | f1ofun 6052 | . . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻) | |
4 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 4 | funimaex 5890 | . . . 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 6491 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵)) |
8 | isocnv 6480 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | |
9 | isof1o 6473 | . . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻:𝐵–1-1-onto→𝐴) | |
10 | f1ofun 6052 | . . . 4 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → Fun ◡𝐻) | |
11 | 4 | funimaex 5890 | . . . 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 6491 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Se 𝐵 → 𝑅 Se 𝐴)) |
14 | 7, 13 | impbid 201 | 1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 Vcvv 3173 Se wse 4995 ◡ccnv 5037 “ cima 5041 Fun wfun 5798 –1-1-onto→wf1o 5803 Isom wiso 5805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-se 4998 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 |
This theorem is referenced by: (None) |
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