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Mirrors > Home > NFE Home > Th. List > coeq1 | GIF version |
Description: Equality theorem for composition of two classes. (Contributed by set.mm contributors, 3-Jan-1997.) |
Ref | Expression |
---|---|
coeq1 | ⊢ (A = B → (A ∘ C) = (B ∘ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coss1 4872 | . . 3 ⊢ (A ⊆ B → (A ∘ C) ⊆ (B ∘ C)) | |
2 | coss1 4872 | . . 3 ⊢ (B ⊆ A → (B ∘ C) ⊆ (A ∘ C)) | |
3 | 1, 2 | anim12i 549 | . 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → ((A ∘ C) ⊆ (B ∘ C) ∧ (B ∘ C) ⊆ (A ∘ C))) |
4 | eqss 3287 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
5 | eqss 3287 | . 2 ⊢ ((A ∘ C) = (B ∘ C) ↔ ((A ∘ C) ⊆ (B ∘ C) ∧ (B ∘ C) ⊆ (A ∘ C))) | |
6 | 3, 4, 5 | 3imtr4i 257 | 1 ⊢ (A = B → (A ∘ C) = (B ∘ C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ⊆ wss 3257 ∘ ccom 4721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 df-br 4640 df-co 4726 |
This theorem is referenced by: coeq1i 4876 coeq1d 4878 composevalg 5817 pprodeq1 5834 pprodeq2 5835 enmap2lem3 6065 enmap2lem5 6067 |
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