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Theorem cofmpt 7131
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 2731 . 2 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3 cofmpt.1 . . 3 (𝜑𝐹:𝐶𝐷)
43feqmptd 6959 . 2 (𝜑𝐹 = (𝑦𝐶 ↦ (𝐹𝑦)))
5 fveq2 6890 . 2 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
61, 2, 4, 5fmptco 7128 1 (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  cmpt 5230  ccom 5679  wf 6538  cfv 6542
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550
This theorem is referenced by:  offsplitfpar  8107  lo1o12  15481  rlimcn1b  15537  mdetralt  22330  tsmsmhm  23870  uniioombllem3  25334  ismbfcn2  25387  itg1climres  25464  iblabslem  25577  iblabs  25578  bddmulibl  25588  limccnp  25640  dvcjbr  25701  dvmptcj  25720  dvef  25732  plypf1  25961  lgamgulmlem2  26770  lgamcvg2  26795  lgseisenlem4  27117  esumcocn  33376  ftc1anclem6  36869  rhmcomulmpl  41426  rhmmpl  41427  selvvvval  41459  evlselv  41461  fundcmpsurbijinjpreimafv  46373  fundcmpsurinjimaid  46377
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