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Theorem cofmpt2 31858
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 7115 . . 3 (𝜑 → (𝑦𝐵𝐶):𝐵𝐸)
3 cofmpt2.3 . . 3 (𝜑𝐹:𝐴𝐵)
4 fcompt 7131 . . 3 (((𝑦𝐵𝐶):𝐵𝐸𝐹:𝐴𝐵) → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))))
52, 3, 4syl2anc 585 . 2 (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))))
6 eqid 2733 . . . 4 (𝑦𝐵𝐶) = (𝑦𝐵𝐶)
7 cofmpt2.1 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)
87adantlr 714 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)
93ffvelcdmda 7087 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
10 cofmpt2.4 . . . . 5 (𝜑𝐷𝑉)
1110adantr 482 . . . 4 ((𝜑𝑥𝐴) → 𝐷𝑉)
126, 8, 9, 11fvmptd2 7007 . . 3 ((𝜑𝑥𝐴) → ((𝑦𝐵𝐶)‘(𝐹𝑥)) = 𝐷)
1312mpteq2dva 5249 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝑦𝐵𝐶)‘(𝐹𝑥))) = (𝑥𝐴𝐷))
145, 13eqtrd 2773 1 (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cmpt 5232  ccom 5681  wf 6540  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by: (None)
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