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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 6534, 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 6531 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶𝐴) |
5 | fco 6534 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐷⟶𝐴) → (𝐹 ∘ 𝐺):𝐷⟶𝐵) | |
6 | 1, 4, 5 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3939 ∘ ccom 5562 ⟶wf 6354 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: volicoff 42287 voliooicof 42288 ovolval2 42933 ovolval5lem2 42942 ovnovollem1 42945 ovnovollem2 42946 |
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