Theorem constcof 32566

Description: Composition with a constant function. See also fcoconst 7068. (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 6712	. . . 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 6689	. . 3 ⊢ (((𝐼 × {𝑌}) Fn 𝐼 ∧ 𝐹:𝑋⟶𝐼) → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
6	3, 4, 5	syl2anc 584	. 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 6923	. . 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 7018	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝐼)
12		fvconst2g 7138	. . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ (𝐹‘𝑥) ∈ 𝐼) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
13	10, 11, 12	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
14	9, 13	eqtrd 2764	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = 𝑌)
15	6, 14	fconst7v 32565	1 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4577   × cxp 5617   ∘ ccom 5623   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490

This theorem is referenced by: (None)