Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cofmpt2 Structured version   Visualization version   GIF version

Theorem cofmpt2 32644
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 7135 . . 3 (𝜑 → (𝑦𝐵𝐶):𝐵𝐸)
3 cofmpt2.3 . . 3 (𝜑𝐹:𝐴𝐵)
4 fcompt 7153 . . 3 (((𝑦𝐵𝐶):𝐵𝐸𝐹:𝐴𝐵) → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))))
52, 3, 4syl2anc 584 . 2 (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))))
6 eqid 2737 . . . 4 (𝑦𝐵𝐶) = (𝑦𝐵𝐶)
7 cofmpt2.1 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)
87adantlr 715 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)
93ffvelcdmda 7104 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
10 cofmpt2.4 . . . . 5 (𝜑𝐷𝑉)
1110adantr 480 . . . 4 ((𝜑𝑥𝐴) → 𝐷𝑉)
126, 8, 9, 11fvmptd2 7024 . . 3 ((𝜑𝑥𝐴) → ((𝑦𝐵𝐶)‘(𝐹𝑥)) = 𝐷)
1312mpteq2dva 5242 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))) = (𝑥𝐴𝐷))
145, 13eqtrd 2777 1 (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cmpt 5225  ccom 5689  wf 6557  cfv 6561
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator