Theorem cfsetsnfsetf 44610

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 484	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐴 ∈ 𝑉)
2	1	adantr 482	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝐴 ∈ 𝑉)
3	2	mptexd 7132	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ V)
4		vex 3441	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
5		feq1 6611	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑔:{𝑌}⟶𝐵))
6		cfsetsnfsetfv.g	. . . . . . . . . . 11 ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
7	4, 5, 6	elab2 3618	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐺 ↔ 𝑔:{𝑌}⟶𝐵)
8	7	biimpi 215	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝐺 → 𝑔:{𝑌}⟶𝐵)
9	8	adantl 483	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑔:{𝑌}⟶𝐵)
10		snidg 4599	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
11	10	adantl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ {𝑌})
12	11	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ {𝑌})
13	9, 12	ffvelcdmd 6994	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑔‘𝑌) ∈ 𝐵)
14	13	adantr 482	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) ∈ 𝐵)
15	14	fmpttd 7021	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵)
16		eqeq2 2748	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑌) → ((𝑔‘𝑌) = 𝑏 ↔ (𝑔‘𝑌) = (𝑔‘𝑌)))
17	16	ralbidv 3171	. . . . . . 7 ⊢ (𝑏 = (𝑔‘𝑌) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
18	17	adantl 483	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑏 = (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
19		eqidd 2737	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) = (𝑔‘𝑌))
20	19	ralrimiva 3140	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌))
21	13, 18, 20	rspcedvd 3568	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏)
22	15, 21	jca 513	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ((𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
23		feq1 6611	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (𝑓:𝐴⟶𝐵 ↔ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵))
24		simpl 484	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
25		eqidd 2737	. . . . . . . . 9 ⊢ (((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑎 = 𝑧) → (𝑔‘𝑌) = (𝑔‘𝑌))
26		simpr 486	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
27		fvexd 6819	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) ∈ V)
28		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑓
29		nfmpt1 5189	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
30	28, 29	nfeq 2918	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
31		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑧 ∈ 𝐴
32	30, 31	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴)
33		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑧
34		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑔‘𝑌)
35	24, 25, 26, 27, 32, 33, 34	fvmptdf 6913	. . . . . . . 8 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑔‘𝑌))
36	35	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑔‘𝑌) = 𝑏))
37	36	ralbidva 3169	. . . . . 6 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
38	37	rexbidv 3172	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
39	23, 38	anbi12d 632	. . . 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 3617	. . 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 2848	. 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 7020	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  {cab 2713  ∀wral 3062  ∃wrex 3071  Vcvv 3437  {csn 4565   ↦ cmpt 5164  ⟶wf 6454  ‘cfv 6458

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466

This theorem is referenced by:  cfsetsnfsetf1  44611  cfsetsnfsetfo  44612