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Theorem cofmpt 7152
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 2738 . 2 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3 cofmpt.1 . . 3 (𝜑𝐹:𝐶𝐷)
43feqmptd 6977 . 2 (𝜑𝐹 = (𝑦𝐶 ↦ (𝐹𝑦)))
5 fveq2 6906 . 2 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
61, 2, 4, 5fmptco 7149 1 (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cmpt 5225  ccom 5689  wf 6557  cfv 6561
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  coof  7721  offsplitfpar  8144  lo1o12  15569  rlimcn1b  15625  rhmcomulmpl  22386  rhmmpl  22387  rhmply1vr1  22391  mdetralt  22614  tsmsmhm  24154  uniioombllem3  25620  ismbfcn2  25673  itg1climres  25749  iblabslem  25863  iblabs  25864  bddmulibl  25874  limccnp  25926  dvcjbr  25987  dvmptcj  26006  dvef  26018  plypf1  26251  lgamgulmlem2  27073  lgamcvg2  27098  lgseisenlem4  27422  gsumwrd2dccat  33070  esumcocn  34081  ftc1anclem6  37705  rhmcomulpsr  42561  rhmpsr  42562  selvvvval  42595  evlselv  42597  fundcmpsurbijinjpreimafv  47394  fundcmpsurinjimaid  47398
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