Theorem cfsetsnfsetf 46973

Description: The mapping of the class of singleton functions into the class of constant functions is a function. (Contributed by AV, 14-Sep-2024.)

Hypotheses

Ref	Expression
cfsetsnfsetfv.f	⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
cfsetsnfsetfv.g	⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
cfsetsnfsetfv.h	⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))

Assertion

Ref	Expression
cfsetsnfsetf	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)

Distinct variable groups:   𝐴,𝑎,𝑔   𝑔,𝐺   𝑔,𝑉   𝑔,𝑌   𝐴,𝑏,𝑓,𝑧   𝑥,𝐵   𝐵,𝑎,𝑏,𝑓   𝑔,𝐹   𝐺,𝑎,𝑏,𝑧   𝑉,𝑎,𝑏,𝑧   𝑌,𝑎,𝑏,𝑓,𝑧   𝑥,𝑌,𝑔   𝑔,𝑏,𝑓,𝑧

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑧,𝑔)   𝐹(𝑥,𝑧,𝑓,𝑎,𝑏)   𝐺(𝑥,𝑓)   𝐻(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑓)

Proof of Theorem cfsetsnfsetf

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐴 ∈ 𝑉)
2	1	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝐴 ∈ 𝑉)
3	2	mptexd 7261	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ V)
4		vex 3492	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
5		feq1 6728	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑔:{𝑌}⟶𝐵))
6		cfsetsnfsetfv.g	. . . . . . . . . . 11 ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
7	4, 5, 6	elab2 3698	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐺 ↔ 𝑔:{𝑌}⟶𝐵)
8	7	biimpi 216	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝐺 → 𝑔:{𝑌}⟶𝐵)
9	8	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑔:{𝑌}⟶𝐵)
10		snidg 4682	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
11	10	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ {𝑌})
12	11	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ {𝑌})
13	9, 12	ffvelcdmd 7119	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑔‘𝑌) ∈ 𝐵)
14	13	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) ∈ 𝐵)
15	14	fmpttd 7149	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵)
16		eqeq2 2752	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑌) → ((𝑔‘𝑌) = 𝑏 ↔ (𝑔‘𝑌) = (𝑔‘𝑌)))
17	16	ralbidv 3184	. . . . . . 7 ⊢ (𝑏 = (𝑔‘𝑌) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
18	17	adantl 481	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑏 = (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
19		eqidd 2741	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) = (𝑔‘𝑌))
20	19	ralrimiva 3152	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌))
21	13, 18, 20	rspcedvd 3637	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏)
22	15, 21	jca 511	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ((𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
23		feq1 6728	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (𝑓:𝐴⟶𝐵 ↔ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵))
24		simpl 482	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
25		eqidd 2741	. . . . . . . . 9 ⊢ (((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑎 = 𝑧) → (𝑔‘𝑌) = (𝑔‘𝑌))
26		simpr 484	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
27		fvexd 6935	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) ∈ V)
28		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑓
29		nfmpt1 5274	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
30	28, 29	nfeq 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
31		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑧 ∈ 𝐴
32	30, 31	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴)
33		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑧
34		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑔‘𝑌)
35	24, 25, 26, 27, 32, 33, 34	fvmptdf 7035	. . . . . . . 8 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑔‘𝑌))
36	35	eqeq1d 2742	. . . . . . 7 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑔‘𝑌) = 𝑏))
37	36	ralbidva 3182	. . . . . 6 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
38	37	rexbidv 3185	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
39	23, 38	anbi12d 631	. . . 4 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → ((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) ↔ ((𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏)))
40	3, 22, 39	elabd 3697	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)})
41		cfsetsnfsetfv.f	. . 3 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
42	40, 41	eleqtrrdi 2855	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ 𝐹)
43		cfsetsnfsetfv.h	. 2 ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
44	42, 43	fmptd 7148	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488  {csn 4648   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  cfsetsnfsetf1  46974  cfsetsnfsetfo  46975