Theorem constcof 32977

Description: Composition with a constant function. See also fcoconst 7130. (Contributed by Thierry Arnoux, 11-Jan-2026.)

Hypotheses

Ref	Expression
constcof.1	⊢ (𝜑 → 𝐹:𝑋⟶𝐼)
constcof.2	⊢ (𝜑 → 𝑌 ∈ 𝑉)

Assertion

Ref	Expression
constcof	⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))

Proof of Theorem constcof

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		constcof.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
2		fnconstg 6766	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (𝐼 × {𝑌}) Fn 𝐼)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → (𝐼 × {𝑌}) Fn 𝐼)
4		constcof.1	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝐼)
5		fnfco 6743	. . 3 ⊢ (((𝐼 × {𝑌}) Fn 𝐼 ∧ 𝐹:𝑋⟶𝐼) → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
6	3, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
7	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝐼)
8		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
9	7, 8	fvco3d 6982	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = ((𝐼 × {𝑌})‘(𝐹‘𝑥)))
10	1	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ 𝑉)
11	4	ffvelcdmda 7079	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝐼)
12		fvconst2g 7200	. . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ (𝐹‘𝑥) ∈ 𝐼) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
13	10, 11, 12	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
14	9, 13	eqtrd 2797	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = 𝑌)
15	6, 14	fconst7v 32976	1 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {csn 4588   × cxp 5658   ∘ ccom 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544

This theorem is used by:  mplvrpmrhm  33946