Theorem constcof 33284

Description: Composition with a constant function. See also fcoconst 7156. (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 6789	. . . 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 6766	. . 3 ⊢ (((𝐼 × {𝑌}) Fn 𝐼 ∧ 𝐹:𝑋⟶𝐼) → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
6	3, 4, 5	syl2anc 596	. 2 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) Fn 𝑋)
7	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝐼)
8		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
9	7, 8	fvco3d 7006	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = ((𝐼 × {𝑌})‘(𝐹‘𝑥)))
10	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ 𝑉)
11	4	ffvelcdmda 7105	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝐼)
12		fvconst2g 7228	. . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ (𝐹‘𝑥) ∈ 𝐼) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
13	10, 11, 12	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐼 × {𝑌})‘(𝐹‘𝑥)) = 𝑌)
14	9, 13	eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐼 × {𝑌}) ∘ 𝐹)‘𝑥) = 𝑌)
15	6, 14	fconst7v 33283	1 ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {csn 4588   × cxp 5653   ∘ ccom 5659   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563

This theorem is used by:  mplvrpmrhm  34246