Theorem fconst7v 32624

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 5720	. . . 4 ⊢ (∅ × {𝐵}) = ∅
2	1	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → (∅ × {𝐵}) = ∅)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅)
4	3	xpeq1d 5650	. . 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 6581	. . . . . 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 6620	. . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
11	9, 10	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = ∅)
12	2, 4, 11	3eqtr4rd 2779	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐹 = (𝐴 × {𝐵}))
13		fconst7v.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
14		fvexd 6846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
15	13, 14	eqeltrrd 2834	. . . . . . . 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 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ {𝐵})
19	18	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})
20		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥𝐴
21		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥{𝐵}
22		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥𝐹
23	20, 21, 22	ffnfvf 7062	. . . . 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 32472	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ V)
29		fconst2g 7146	. . . 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 2939	. . 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 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  Vcvv 3437  ∅c0 4282  {csn 4577   × cxp 5619   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  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 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497

This theorem is referenced by:  constcof  32625  extdgfialglem2  33778