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Mirrors > Home > MPE Home > Th. List > coeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.) |
Ref | Expression |
---|---|
coeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coss1 5199 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶)) | |
2 | coss1 5199 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∘ 𝐶) ⊆ (𝐴 ∘ 𝐶)) | |
3 | 1, 2 | anim12i 588 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → ((𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶) ∧ (𝐵 ∘ 𝐶) ⊆ (𝐴 ∘ 𝐶))) |
4 | eqss 3583 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 3583 | . 2 ⊢ ((𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) ↔ ((𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶) ∧ (𝐵 ∘ 𝐶) ⊆ (𝐴 ∘ 𝐶))) | |
6 | 3, 4, 5 | 3imtr4i 280 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ⊆ wss 3540 ∘ ccom 5042 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-co 5047 |
This theorem is referenced by: coeq1i 5203 coeq1d 5205 coi2 5569 relcnvtr 5572 funcoeqres 6080 ereq1 7636 domssex2 8005 wemapwe 8477 seqf1olem2 12703 seqf1o 12704 relexpsucnnl 13620 isps 17025 pwsco1mhm 17193 frmdup3 17227 symgov 17633 pmtr3ncom 17718 psgnunilem1 17736 frgpup3 18014 gsumval3 18131 evlseu 19337 evlsval2 19341 evls1val 19506 evls1sca 19509 evl1val 19514 mpfpf1 19536 pf1mpf 19537 pf1ind 19540 frgpcyg 19741 frlmup4 19959 xkococnlem 21272 xkococn 21273 cnmpt1k 21295 cnmptkk 21296 xkofvcn 21297 qtopeu 21329 qtophmeo 21430 utop2nei 21864 cncombf 23231 dgrcolem2 23834 dgrco 23835 motplusg 25237 hocsubdir 28028 hoddi 28233 opsqrlem1 28383 smatfval 29189 msubco 30682 trljco 35046 tgrpov 35054 tendovalco 35071 erngmul 35112 erngmul-rN 35120 cdlemksv 35150 cdlemkuu 35201 cdlemk41 35226 cdleml5N 35286 cdleml9 35290 dvamulr 35318 dvavadd 35321 dvhmulr 35393 dvhvscacbv 35405 dvhvscaval 35406 dih1dimatlem0 35635 dihjatcclem4 35728 diophrw 36340 eldioph2 36343 diophren 36395 mendmulr 36777 rngcinv 41773 rngcinvALTV 41785 ringcinv 41824 ringcinvALTV 41848 |
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