Theorem cofmpt 7133

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 2762	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
3		cofmpt.1	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷)
4	3	feqmptd 6953	. 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦)))
5		fveq2 6885	. 2 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
6	1, 2, 4, 5	fmptco 7130	1 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5186   ∘ ccom 5655  ⟶wf 6534  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546

This theorem is used by:  coof  7717  offsplitfpar  8130  lo1o12  15700  rlimcn1b  15756  rhmcomulmpl  22433  selvvvval  22451  rhmmpl  22698  rhmply1vr1  22702  mdetralt  22923  tsmsmhm  24465  uniioombllem3  25906  ismbfcn2  25959  itg1climres  26035  iblabslem  26148  iblabs  26149  bddmulibl  26159  limccnp  26211  dvcjbr  26269  dvmptcj  26288  dvef  26300  plypf1  26531  lgamgulmlem2  27357  lgamcvg2  27382  lgseisenlem4  27705  gsummulsubdishift2  33630  gsumwrd2dccat  33639  selvply1rhmlema  34150  selvply1rhmlem1  34152  extvfvcl  34168  esumcocn  34712  ftc1anclem6  38616  rhmcomulpsr  43610  rhmpsr  43611  evlselv  43617  fundcmpsurbijinjpreimafv  48488  fundcmpsurinjimaid  48492