Theorem cofmpt 7085

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 2737	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
3		cofmpt.1	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷)
4	3	feqmptd 6908	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
5		fveq2 6840	. 2 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
6	1, 2, 4, 5	fmptco 7082	1 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5166   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by:  coof  7655  offsplitfpar  8069  lo1o12  15495  rlimcn1b  15551  rhmcomulmpl  22347  rhmmpl  22348  rhmply1vr1  22352  mdetralt  22573  tsmsmhm  24111  uniioombllem3  25552  ismbfcn2  25605  itg1climres  25681  iblabslem  25795  iblabs  25796  bddmulibl  25806  limccnp  25858  dvcjbr  25916  dvmptcj  25935  dvef  25947  plypf1  26177  lgamgulmlem2  26993  lgamcvg2  27018  lgseisenlem4  27341  gsummulsubdishift2  33130  gsumwrd2dccat  33139  extvfvcl  33680  esumcocn  34224  ftc1anclem6  38019  rhmcomulpsr  42994  rhmpsr  42995  selvvvval  43018  evlselv  43020  fundcmpsurbijinjpreimafv  47867  fundcmpsurinjimaid  47871