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Mirrors > Home > MPE Home > Th. List > fco2 | Structured version Visualization version GIF version |
Description: Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.) |
Ref | Expression |
---|---|
fco2 | ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fco 6741 | . 2 ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → ((𝐹 ↾ 𝐵) ∘ 𝐺):𝐴⟶𝐶) | |
2 | frn 6724 | . . . . 5 ⊢ (𝐺:𝐴⟶𝐵 → ran 𝐺 ⊆ 𝐵) | |
3 | cores 6248 | . . . . 5 ⊢ (ran 𝐺 ⊆ 𝐵 → ((𝐹 ↾ 𝐵) ∘ 𝐺) = (𝐹 ∘ 𝐺)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐺:𝐴⟶𝐵 → ((𝐹 ↾ 𝐵) ∘ 𝐺) = (𝐹 ∘ 𝐺)) |
5 | 4 | adantl 481 | . . 3 ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → ((𝐹 ↾ 𝐵) ∘ 𝐺) = (𝐹 ∘ 𝐺)) |
6 | 5 | feq1d 6702 | . 2 ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (((𝐹 ↾ 𝐵) ∘ 𝐺):𝐴⟶𝐶 ↔ (𝐹 ∘ 𝐺):𝐴⟶𝐶)) |
7 | 1, 6 | mpbid 231 | 1 ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊆ wss 3948 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 |
This theorem is referenced by: fsuppcor 9405 prdsrngd 20077 prdsringd 20216 prdscrngd 20217 prds1 20218 prdstmdd 23948 prdsxmslem2 24358 eulerpartlemmf 33838 sseqf 33855 poimirlem9 36961 ftc1anclem3 37027 fco2d 43377 |
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