Theorem constcof 32713

Description: Composition with a constant function. See also fcoconst 7083. (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 6724	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (𝐼 × {𝑌}) Fn 𝐼)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (𝐼 × {𝑌}) Fn 𝐼)
4		constcof.1	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝐼)
5		fnfco 6701	. . 3 ⊢ (((𝐼 × {𝑌}) Fn 𝐼 ∧ 𝐹:𝑋⟶𝐼) → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
6	3, 4, 5	syl2anc 585	. 2 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
7	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝐼)
8		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
9	7, 8	fvco3d 6936	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = ((𝐼 × {𝑌})‘(𝐹‘𝑥)))
10	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ 𝑉)
11	4	ffvelcdmda 7032	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝐼)
12		fvconst2g 7152	. . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ (𝐹‘𝑥) ∈ 𝐼) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
13	10, 11, 12	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
14	9, 13	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = 𝑌)
15	6, 14	fconst7v 32712	1 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4568   × cxp 5624   ∘ ccom 5630   Fn wfn 6489  ⟶wf 6490  ‘cfv 6494

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 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502

This theorem is referenced by:  mplvrpmrhm  33710