Theorem cfsetsnfsetf 47528

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 483	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐴 ∈ 𝑉)
2	1	adantr 481	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝐴 ∈ 𝑉)
3	2	mptexd 7175	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ V)
4		vex 3436	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
5		feq1 6640	. . . . . . . . . 10 ⊢ (𝑥 = 𝑔 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑔:{𝑌}⟶𝐵))
6		cfsetsnfsetfv.g	. . . . . . . . . 10 ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
7	4, 5, 6	elab2 3627	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝐺 ↔ 𝑔:{𝑌}⟶𝐵)
8	7	bilani 505	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑔:{𝑌}⟶𝐵)
9		snidg 4599	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
10	9	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ {𝑌})
11	10	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ {𝑌})
12	8, 11	ffvelcdmd 7033	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑔‘𝑌) ∈ 𝐵)
13	12	adantr 481	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) ∈ 𝐵)
14	13	fmpttd 7063	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵)
15		eqeq2 2752	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑌) → ((𝑔‘𝑌) = 𝑏 ↔ (𝑔‘𝑌) = (𝑔‘𝑌)))
16	15	ralbidv 3163	. . . . . . 7 ⊢ (𝑏 = (𝑔‘𝑌) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
17	16	adantl 482	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑏 = (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
18		eqidd 2741	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) = (𝑔‘𝑌))
19	18	ralrimiva 3132	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌))
20	12, 17, 19	rspcedvd 3569	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏)
21	14, 20	jca 516	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ((𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
22		feq1 6640	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (𝑓:𝐴⟶𝐵 ↔ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵))
23		simpl 483	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
24		eqidd 2741	. . . . . . . . 9 ⊢ (((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑎 = 𝑧) → (𝑔‘𝑌) = (𝑔‘𝑌))
25		simpr 485	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
26		fvexd 6849	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) ∈ V)
27		nfcv 2902	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑓
28		nfmpt1 5178	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
29	27, 28	nfeq 2915	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
30		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑧 ∈ 𝐴
31	29, 30	nfan 1906	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴)
32		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑧
33		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑔‘𝑌)
34	23, 24, 25, 26, 31, 32, 33	fvmptdf 6949	. . . . . . . 8 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑔‘𝑌))
35	34	eqeq1d 2742	. . . . . . 7 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑔‘𝑌) = 𝑏))
36	35	ralbidva 3161	. . . . . 6 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
37	36	rexbidv 3164	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
38	22, 37	anbi12d 638	. . . 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)))
39	3, 21, 38	elabd 3626	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)})
40		cfsetsnfsetfv.f	. . 3 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
41	39, 40	eleqtrrdi 2851	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ 𝐹)
42		cfsetsnfsetfv.h	. 2 ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
43	41, 42	fmptd 7062	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064  Vcvv 3432  {csn 4562   ↦ cmpt 5160  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  cfsetsnfsetf1  47529  cfsetsnfsetfo  47530