Theorem fconst7v 32584

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 5722	. . . 4 ⊢ (∅ × {𝐵}) = ∅
2	1	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (∅ × {𝐵}) = ∅)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅)
4	3	xpeq1d 5652	. . 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 6578	. . . . . 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 6617	. . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
11	9, 10	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = ∅)
12	2, 4, 11	3eqtr4rd 2775	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = (𝐴 × {𝐵}))
13		fconst7v.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
14		fvexd 6841	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
15	13, 14	eqeltrrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
16		snidg 4614	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵})
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝐵})
18	13, 17	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ {𝐵})
19	18	ralrimiva 3121	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})
20		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝐴
21		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥{𝐵}
22		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝐹
23	20, 21, 22	ffnfvf 7058	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵}))
24	5, 19, 23	sylanbrc 583	. . . 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 715	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
28	26, 27	n0limd 32435	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V)
29		fconst2g 7143	. . . 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 2935	. . 3 ⊢ (𝐴 = ∅ ∨ 𝐴 ≠ ∅)
33	32	a1i 11	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 ≠ ∅))
34	12, 31, 33	mpjaodan 960	1 ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3438  ∅c0 4286  {csn 4579   × cxp 5621   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486

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 5238  ax-nul 5248  ax-pr 5374

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 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494

This theorem is referenced by:  extdgfialglem2  33679