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Theorem fcomptss 38390
 Description: Express composition of two functions as a maps-to applying both in sequence. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
fcomptss.a (𝜑𝐹:𝐴𝐵)
fcomptss.b (𝜑𝐵𝐶)
fcomptss.g (𝜑𝐺:𝐶𝐷)
Assertion
Ref Expression
fcomptss (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐹   𝑥,𝐺
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fcomptss
StepHypRef Expression
1 fcomptss.g . 2 (𝜑𝐺:𝐶𝐷)
2 fcomptss.a . . 3 (𝜑𝐹:𝐴𝐵)
3 fcomptss.b . . 3 (𝜑𝐵𝐶)
42, 3fssd 5970 . 2 (𝜑𝐹:𝐴𝐶)
5 fcompt 6306 . 2 ((𝐺:𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
61, 4, 5syl2anc 691 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ⊆ wss 3540   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  ovolval2lem  39533  ovolval5lem2  39543
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