Theorem fconst7v 32710

Description: An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) Removed hyphotheses as suggested by SN (Revised by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
fconst7v.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
fconst7v.e	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Assertion

Ref	Expression
fconst7v	⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝜑,𝑥

Proof of Theorem fconst7v

Step	Hyp	Ref	Expression
1		0xp 5731	. . . 4 ⊢ (∅ × {𝐵}) = ∅
2	1	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (∅ × {𝐵}) = ∅)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅)
4	3	xpeq1d 5661	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (𝐴 × {𝐵}) = (∅ × {𝐵}))
5		fconst7v.f	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
7		fneq2 6592	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐹 Fn 𝐴 ↔ 𝐹 Fn ∅))
8	7	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ 𝐹 Fn ∅))
9	6, 8	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
10		fn0 6631	. . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
11	9, 10	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = ∅)
12	2, 4, 11	3eqtr4rd 2783	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = (𝐴 × {𝐵}))
13		fconst7v.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
14		fvexd 6857	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
15	13, 14	eqeltrrd 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
16		snidg 4619	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵})
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝐵})
18	13, 17	eqeltrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ {𝐵})
19	18	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})
20		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝐴
21		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥{𝐵}
22		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝐹
23	20, 21, 22	ffnfvf 7074	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵}))
24	5, 19, 23	sylanbrc 584	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶{𝐵})
25	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐹:𝐴⟶{𝐵})
26		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
27	15	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
28	26, 27	n0limd 32558	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V)
29		fconst2g 7159	. . . 4 ⊢ (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
30	28, 29	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
31	25, 30	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐹 = (𝐴 × {𝐵}))
32		exmidne 2943	. . 3 ⊢ (𝐴 = ∅ ∨ 𝐴 ≠ ∅)
33	32	a1i 11	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 ≠ ∅))
34	12, 31, 33	mpjaodan 961	1 ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442  ∅c0 4287  {csn 4582   × cxp 5630   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500

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 5243  ax-nul 5253  ax-pr 5379

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 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508

This theorem is referenced by:  constcof  32711  extdgfialglem2  33871