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Theorem cofmpt 6871
Description: Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
Hypotheses
Ref Expression
cofmpt.1 (𝜑𝐹:𝐶𝐷)
cofmpt.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
cofmpt (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem cofmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cofmpt.2 . 2 ((𝜑𝑥𝐴) → 𝐵𝐶)
2 eqidd 2799 . 2 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3 cofmpt.1 . . 3 (𝜑𝐹:𝐶𝐷)
43feqmptd 6708 . 2 (𝜑𝐹 = (𝑦𝐶 ↦ (𝐹𝑦)))
5 fveq2 6645 . 2 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
61, 2, 4, 5fmptco 6868 1 (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  cmpt 5110  ccom 5523  wf 6320  cfv 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332
This theorem is referenced by:  offsplitfpar  7798  lo1o12  14882  rlimcn1b  14938  mdetralt  21213  tsmsmhm  22751  uniioombllem3  24189  ismbfcn2  24242  itg1climres  24318  iblabslem  24431  iblabs  24432  bddmulibl  24442  limccnp  24494  dvcjbr  24552  dvmptcj  24571  dvef  24583  plypf1  24809  lgamgulmlem2  25615  lgamcvg2  25640  lgseisenlem4  25962  esumcocn  31449  ftc1anclem6  35135  fundcmpsurbijinjpreimafv  43922  fundcmpsurinjimaid  43926
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