Theorem constcof 32717

Description: Composition with a constant function. See also fcoconst 7080. (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 6719	. . . 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 6696	. . 3 ⊢ (((𝐼 × {𝑌}) Fn 𝐼 ∧ 𝐹:𝑋⟶𝐼) → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
6	3, 4, 5	syl2anc 591	. 2 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
7	4	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝐼)
8		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
9	7, 8	fvco3d 6932	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = ((𝐼 × {𝑌})‘(𝐹‘𝑥)))
10	1	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ 𝑉)
11	4	ffvelcdmda 7029	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝐼)
12		fvconst2g 7150	. . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ (𝐹‘𝑥) ∈ 𝐼) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
13	10, 11, 12	syl2anc 591	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
14	9, 13	eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = 𝑌)
15	6, 14	fconst7v 32716	1 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {csn 4558   × cxp 5619   ∘ ccom 5625   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497

This theorem is referenced by:  mplvrpmrhm  33743