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Theorem fcoss 44454
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.)
Hypotheses
Ref Expression
fcoss.f (𝜑𝐹:𝐴𝐵)
fcoss.c (𝜑𝐶𝐴)
fcoss.g (𝜑𝐺:𝐷𝐶)
Assertion
Ref Expression
fcoss (𝜑 → (𝐹𝐺):𝐷𝐵)

Proof of Theorem fcoss
StepHypRef Expression
1 fcoss.f . 2 (𝜑𝐹:𝐴𝐵)
2 fcoss.g . . 3 (𝜑𝐺:𝐷𝐶)
3 fcoss.c . . 3 (𝜑𝐶𝐴)
42, 3fssd 6726 . 2 (𝜑𝐺:𝐷𝐴)
5 fco 6732 . 2 ((𝐹:𝐴𝐵𝐺:𝐷𝐴) → (𝐹𝐺):𝐷𝐵)
61, 4, 5syl2anc 583 1 (𝜑 → (𝐹𝐺):𝐷𝐵)
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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