Theorem cfsetsnfsetf 48343

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 488	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐴 ∈ 𝑉)
2	1	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝐴 ∈ 𝑉)
3	2	mptexd 7256	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∈ V)
4		vex 3458	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
5		feq1 6712	. . . . . . . . . 10 ⊢ (𝑥 = 𝑔 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑔:{𝑌}⟶𝐵))
6		cfsetsnfsetfv.g	. . . . . . . . . 10 ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
7	4, 5, 6	elab2 3639	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝐺 ↔ 𝑔:{𝑌}⟶𝐵)
8	7	bilani 510	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑔:{𝑌}⟶𝐵)
9		snidg 4621	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
10	9	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ {𝑌})
11	10	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ {𝑌})
12	8, 11	ffvelcdmd 7112	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑔‘𝑌) ∈ 𝐵)
13	12	adantr 486	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) ∈ 𝐵)
14	13	fmpttd 7142	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵)
15		eqeq2 2775	. . . . . . . 8 ⊢ (𝑏 = (𝑔‘𝑌) → ((𝑔‘𝑌) = 𝑏 ↔ (𝑔‘𝑌) = (𝑔‘𝑌)))
16	15	ralbidv 3189	. . . . . . 7 ⊢ (𝑏 = (𝑔‘𝑌) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
17	16	adantl 487	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑏 = (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌)))
18		eqidd 2764	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) = (𝑔‘𝑌))
19	18	ralrimiva 3158	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = (𝑔‘𝑌))
20	12, 17, 19	rspcedvd 3582	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏)
21	14, 20	jca 521	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑔 ∈ 𝐺) → ((𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
22		feq1 6712	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (𝑓:𝐴⟶𝐵 ↔ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)):𝐴⟶𝐵))
23		simpl 488	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
24		eqidd 2764	. . . . . . . . 9 ⊢ (((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑎 = 𝑧) → (𝑔‘𝑌) = (𝑔‘𝑌))
25		simpr 490	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
26		fvexd 6926	. . . . . . . . 9 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑌) ∈ V)
27		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑓
28		nfmpt1 5208	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
29	27, 28	nfeq 2939	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))
30		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑎 𝑧 ∈ 𝐴
31	29, 30	nfan 1934	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴)
32		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑧
33		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑔‘𝑌)
34	23, 24, 25, 26, 31, 32, 33	fvmptdf 7026	. . . . . . . 8 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑔‘𝑌))
35	34	eqeq1d 2765	. . . . . . 7 ⊢ ((𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑔‘𝑌) = 𝑏))
36	35	ralbidva 3187	. . . . . 6 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
37	36	rexbidv 3190	. . . . 5 ⊢ (𝑓 = (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑔‘𝑌) = 𝑏))
38	22, 37	anbi12d 644	. . . 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 3638	. . 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 2875	. 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 7141	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454  {csn 4588   ↦ cmpt 5190  ⟶wf 6557  ‘cfv 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569

This theorem is used by:  cfsetsnfsetf1  48344  cfsetsnfsetfo  48345