Theorem cofmpt 7132

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	⊢ (𝜑 → 𝐹:𝐶⟶𝐷)
cofmpt.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
cofmpt	⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)))

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.

Step	Hyp	Ref	Expression
1		cofmpt.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
2		eqidd 2735	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
3		cofmpt.1	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷)
4	3	feqmptd 6957	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
5		fveq2 6886	. 2 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
6	1, 2, 4, 5	fmptco 7129	1 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ↦ cmpt 5205   ∘ ccom 5669  ⟶wf 6537  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549

This theorem is referenced by:  coof  7703  offsplitfpar  8126  lo1o12  15552  rlimcn1b  15608  rhmcomulmpl  22335  rhmmpl  22336  rhmply1vr1  22340  mdetralt  22563  tsmsmhm  24101  uniioombllem3  25557  ismbfcn2  25610  itg1climres  25686  iblabslem  25800  iblabs  25801  bddmulibl  25811  limccnp  25863  dvcjbr  25924  dvmptcj  25943  dvef  25955  plypf1  26188  lgamgulmlem2  27010  lgamcvg2  27035  lgseisenlem4  27359  gsumwrd2dccat  33014  esumcocn  34056  ftc1anclem6  37680  rhmcomulpsr  42540  rhmpsr  42541  selvvvval  42574  evlselv  42576  fundcmpsurbijinjpreimafv  47367  fundcmpsurinjimaid  47371