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| Mirrors > Home > MPE Home > Th. List > coeq2i | Structured version Visualization version GIF version | ||
| Description: Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
| Ref | Expression |
|---|---|
| coeq1i.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| coeq2i | ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | coeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | coeq2 5869 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-br 5144 df-opab 5206 df-co 5694 |
| This theorem is referenced by: coeq12i 5874 cocnvcnv2 6278 co01 6281 dfpo2 6316 fcoi1 6782 f1ofvswap 7326 dftpos2 8268 tposco 8282 cottrcl 9759 canthp1 10694 cats1co 14895 isoval 17809 mvdco 19463 evlsval 22110 evl1fval1lem 22334 evl1var 22340 pf1ind 22359 rhmply1vr1 22391 rhmply1vsca 22392 imasdsf1olem 24383 hoico1 31775 hoid1i 31808 pjclem1 32214 pjclem3 32216 pjci 32219 cycpmconjv 33162 cycpmconjs 33176 poimirlem9 37636 cdlemk45 40949 cononrel1 43607 trclubgNEW 43631 trclrelexplem 43724 relexpaddss 43731 cotrcltrcl 43738 cortrcltrcl 43753 corclrtrcl 43754 cotrclrcl 43755 cortrclrcl 43756 cotrclrtrcl 43757 cortrclrtrcl 43758 brco3f1o 44046 clsneibex 44115 neicvgbex 44125 subsaliuncl 46373 meadjiun 46481 fundcmpsurinjimaid 47398 dftpos5 48774 tposrescnv 48779 |
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