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Theorem cofmpt2 30491
 Description: Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 15-Jul-2023.)
Hypotheses
Ref Expression
cofmpt2.1 ((𝜑𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)
cofmpt2.2 ((𝜑𝑦𝐵) → 𝐶𝐸)
cofmpt2.3 (𝜑𝐹:𝐴𝐵)
cofmpt2.4 (𝜑𝐷𝑉)
Assertion
Ref Expression
cofmpt2 (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴𝐷))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶   𝑦,𝐷   𝑥,𝐸,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem cofmpt2
StepHypRef Expression
1 cofmpt2.2 . . . 4 ((𝜑𝑦𝐵) → 𝐶𝐸)
21fmpttd 6870 . . 3 (𝜑 → (𝑦𝐵𝐶):𝐵𝐸)
3 cofmpt2.3 . . 3 (𝜑𝐹:𝐴𝐵)
4 fcompt 6886 . . 3 (((𝑦𝐵𝐶):𝐵𝐸𝐹:𝐴𝐵) → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))))
52, 3, 4syl2anc 587 . 2 (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))))
6 eqid 2758 . . . 4 (𝑦𝐵𝐶) = (𝑦𝐵𝐶)
7 cofmpt2.1 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)
87adantlr 714 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)
93ffvelrnda 6842 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
10 cofmpt2.4 . . . . 5 (𝜑𝐷𝑉)
1110adantr 484 . . . 4 ((𝜑𝑥𝐴) → 𝐷𝑉)
126, 8, 9, 11fvmptd2 6767 . . 3 ((𝜑𝑥𝐴) → ((𝑦𝐵𝐶)‘(𝐹𝑥)) = 𝐷)
1312mpteq2dva 5127 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))) = (𝑥𝐴𝐷))
145, 13eqtrd 2793 1 (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5112   ∘ ccom 5528  ⟶wf 6331  ‘cfv 6335 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343 This theorem is referenced by: (None)
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