Theorem fconst7v 32905

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 5761	. . . 4 ⊢ (∅ × {𝐵}) = ∅
2	1	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (∅ × {𝐵}) = ∅)
3		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅)
4	3	xpeq1d 5691	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (𝐴 × {𝐵}) = (∅ × {𝐵}))
5		fconst7v.f	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6	5	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
7		fneq2 6628	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐹 Fn 𝐴 ↔ 𝐹 Fn ∅))
8	7	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ 𝐹 Fn ∅))
9	6, 8	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
10		fn0 6667	. . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
11	9, 10	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = ∅)
12	2, 4, 11	3eqtr4rd 2815	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = (𝐴 × {𝐵}))
13		fconst7v.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
14		fvexd 6897	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
15	13, 14	eqeltrrd 2870	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
16		snidg 4631	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵})
17	15, 16	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝐵})
18	13, 17	eqeltrd 2869	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ {𝐵})
19	18	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})
20		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑥𝐴
21		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑥{𝐵}
22		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑥𝐹
23	20, 21, 22	ffnfvf 7116	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵}))
24	5, 19, 23	sylanbrc 594	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶{𝐵})
25	24	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐹:𝐴⟶{𝐵})
26		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
27	15	adantlr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
28	26, 27	n0limd 4316	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V)
29		fconst2g 7202	. . . 4 ⊢ (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
30	28, 29	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
31	25, 30	mpbid 235	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐹 = (𝐴 × {𝐵}))
32		exmidne 2974	. . 3 ⊢ (𝐴 = ∅ ∨ 𝐴 ≠ ∅)
33	32	a1i 11	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 ≠ ∅))
34	12, 31, 33	mpjaodan 973	1 ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  Vcvv 3463  ∅c0 4294  {csn 4594   × cxp 5660   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is referenced by:  constcof  32906  extdgfialglem2  34027