Theorem cofmpt2 32707

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 7067	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐸)
3		cofmpt2.3	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4		fcompt 7086	. . 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 585	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))))
6		eqid 2736	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶)
7		cofmpt2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)
8	7	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)
9	3	ffvelcdmda 7036	. . . 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 6956	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)) = 𝐷)
13	12	mpteq2dva 5178	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝐷))
14	5, 13	eqtrd 2771	1 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5166   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by: (None)