Theorem cofmpt2 30379

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 6876	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐸)
3		cofmpt2.3	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4		fcompt 6892	. . 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 586	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))))
6		eqid 2820	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶)
7		cofmpt2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)
8	7	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)
9	3	ffvelrnda 6848	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
10		cofmpt2.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
11	10	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝑉)
12	6, 8, 9, 11	fvmptd2 6773	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)) = 𝐷)
13	12	mpteq2dva 5158	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝐷))
14	5, 13	eqtrd 2855	1 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ↦ cmpt 5143   ∘ ccom 5556  ⟶wf 6348  ‘cfv 6352

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3495  df-sbc 3771  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5457  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360

This theorem is referenced by: (None)