Theorem cfsetsnfsetf 47183

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 7164	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ V)
4		vex 3441	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
5		feq1 6634	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑔:{𝑌}⟶𝐵))
6		cfsetsnfsetfv.g	. . . . . . . . . . 11 ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
7	4, 5, 6	elab2 3634	. . . . . . . . . 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 4612	. . . . . . . . . 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 7024	. . . . . . 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 7054	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵)
16		eqeq2 2745	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑌) → ((𝑔‘𝑌) = 𝑏 ↔ (𝑔‘𝑌) = (𝑔‘𝑌)))
17	16	ralbidv 3156	. . . . . . 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 2734	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) = (𝑔‘𝑌))
20	19	ralrimiva 3125	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌))
21	13, 18, 20	rspcedvd 3575	. . . . 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 6634	. . . . 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 2734	. . . . . . . . 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 6843	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) ∈ V)
28		nfcv 2895	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑓
29		nfmpt1 5192	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
30	28, 29	nfeq 2909	. . . . . . . . . 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 2895	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑧
34		nfcv 2895	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑔‘𝑌)
35	24, 25, 26, 27, 32, 33, 34	fvmptdf 6941	. . . . . . . 8 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑔‘𝑌))
36	35	eqeq1d 2735	. . . . . . 7 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑔‘𝑌) = 𝑏))
37	36	ralbidva 3154	. . . . . 6 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
38	37	rexbidv 3157	. . . . 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 3633	. . 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 2844	. 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 7053	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  ∃wrex 3057  Vcvv 3437  {csn 4575   ↦ cmpt 5174  ⟶wf 6482  ‘cfv 6486

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-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

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-reu 3348  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 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494

This theorem is referenced by:  cfsetsnfsetf1  47184  cfsetsnfsetfo  47185