Theorem cofmpt2 32650

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	⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)
cofmpt2.2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐸)
cofmpt2.3	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
cofmpt2.4	⊢ (𝜑 → 𝐷 ∈ 𝑉)

Assertion

Ref	Expression
cofmpt2	⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷))

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

Step	Hyp	Ref	Expression
1		cofmpt2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐸)
2	1	fmpttd 7134	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐸)
3		cofmpt2.3	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4		fcompt 7152	. . 3 ⊢ (((𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐸 ∧ 𝐹:𝐴⟶𝐵) → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))))
5	2, 3, 4	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))))
6		eqid 2734	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶)
7		cofmpt2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)
8	7	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)
9	3	ffvelcdmda 7103	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
10		cofmpt2.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝑉)
12	6, 8, 9, 11	fvmptd2 7023	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)) = 𝐷)
13	12	mpteq2dva 5247	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝐷))
14	5, 13	eqtrd 2774	1 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ↦ cmpt 5230   ∘ ccom 5692  ⟶wf 6558  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570

This theorem is referenced by: (None)