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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoss | Structured version Visualization version GIF version |
Description: Composition of two mappings. Similar to fco 6732, but with a weaker condition on the domain of 𝐹. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
fcoss.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fcoss.c | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
fcoss.g | ⊢ (𝜑 → 𝐺:𝐷⟶𝐶) |
Ref | Expression |
---|---|
fcoss | ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcoss.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | fcoss.g | . . 3 ⊢ (𝜑 → 𝐺:𝐷⟶𝐶) | |
3 | fcoss.c | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
4 | 2, 3 | fssd 6726 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶𝐴) |
5 | fco 6732 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐷⟶𝐴) → (𝐹 ∘ 𝐺):𝐷⟶𝐵) | |
6 | 1, 4, 5 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3941 ∘ ccom 5671 ⟶wf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-fun 6536 df-fn 6537 df-f 6538 |
This theorem is referenced by: volicoff 45256 voliooicof 45257 ovolval2 45905 ovolval5lem2 45914 ovolval5lem3 45915 ovnovollem1 45917 ovnovollem2 45918 |
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