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Theorem fcomptss 44606
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 6745 . 2 (𝜑𝐹:𝐴𝐶)
5 fcompt 7148 . 2 ((𝐺:𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
61, 4, 5syl2anc 582 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wss 3949  cmpt 5235  ccom 5686  wf 6549  cfv 6553
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561
This theorem is referenced by:  ovolval2lem  46060  ovolval5lem2  46070
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