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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 6573 | . . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵) | |
3 | f1ofun 6139 | . . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻) | |
4 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 4 | funimaex 5976 | . . . 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 6591 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 → 𝑆 Se 𝐵)) |
8 | isocnv 6580 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | |
9 | isof1o 6573 | . . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻:𝐵–1-1-onto→𝐴) | |
10 | f1ofun 6139 | . . . 4 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → Fun ◡𝐻) | |
11 | 4 | funimaex 5976 | . . . 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 6591 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Se 𝐵 → 𝑅 Se 𝐴)) |
14 | 7, 13 | impbid 202 | 1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 Vcvv 3200 Se wse 5071 ◡ccnv 5113 “ cima 5117 Fun wfun 5882 –1-1-onto→wf1o 5887 Isom wiso 5889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-se 5074 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 |
This theorem is referenced by: (None) |
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