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Theorem cfsetsnfsetf 46353
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:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
cfsetsnfsetfv.g 𝐺 = {𝑥𝑥:{𝑌}⟶𝐵}
cfsetsnfsetfv.h 𝐻 = (𝑔𝐺 ↦ (𝑎𝐴 ↦ (𝑔𝑌)))
Assertion
Ref Expression
cfsetsnfsetf ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺𝐹)
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
StepHypRef Expression
1 simpl 482 . . . . . 6 ((𝐴𝑉𝑌𝐴) → 𝐴𝑉)
21adantr 480 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → 𝐴𝑉)
32mptexd 7230 . . . 4 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)) ∈ V)
4 vex 3473 . . . . . . . . . . 11 𝑔 ∈ V
5 feq1 6697 . . . . . . . . . . 11 (𝑥 = 𝑔 → (𝑥:{𝑌}⟶𝐵𝑔:{𝑌}⟶𝐵))
6 cfsetsnfsetfv.g . . . . . . . . . . 11 𝐺 = {𝑥𝑥:{𝑌}⟶𝐵}
74, 5, 6elab2 3669 . . . . . . . . . 10 (𝑔𝐺𝑔:{𝑌}⟶𝐵)
87biimpi 215 . . . . . . . . 9 (𝑔𝐺𝑔:{𝑌}⟶𝐵)
98adantl 481 . . . . . . . 8 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → 𝑔:{𝑌}⟶𝐵)
10 snidg 4658 . . . . . . . . . 10 (𝑌𝐴𝑌 ∈ {𝑌})
1110adantl 481 . . . . . . . . 9 ((𝐴𝑉𝑌𝐴) → 𝑌 ∈ {𝑌})
1211adantr 480 . . . . . . . 8 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → 𝑌 ∈ {𝑌})
139, 12ffvelcdmd 7089 . . . . . . 7 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑔𝑌) ∈ 𝐵)
1413adantr 480 . . . . . 6 ((((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) ∧ 𝑎𝐴) → (𝑔𝑌) ∈ 𝐵)
1514fmpttd 7119 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵)
16 eqeq2 2739 . . . . . . . 8 (𝑏 = (𝑔𝑌) → ((𝑔𝑌) = 𝑏 ↔ (𝑔𝑌) = (𝑔𝑌)))
1716ralbidv 3172 . . . . . . 7 (𝑏 = (𝑔𝑌) → (∀𝑧𝐴 (𝑔𝑌) = 𝑏 ↔ ∀𝑧𝐴 (𝑔𝑌) = (𝑔𝑌)))
1817adantl 481 . . . . . 6 ((((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) ∧ 𝑏 = (𝑔𝑌)) → (∀𝑧𝐴 (𝑔𝑌) = 𝑏 ↔ ∀𝑧𝐴 (𝑔𝑌) = (𝑔𝑌)))
19 eqidd 2728 . . . . . . 7 ((((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) ∧ 𝑧𝐴) → (𝑔𝑌) = (𝑔𝑌))
2019ralrimiva 3141 . . . . . 6 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → ∀𝑧𝐴 (𝑔𝑌) = (𝑔𝑌))
2113, 18, 20rspcedvd 3609 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏)
2215, 21jca 511 . . . 4 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → ((𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏))
23 feq1 6697 . . . . 5 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → (𝑓:𝐴𝐵 ↔ (𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵))
24 simpl 482 . . . . . . . . 9 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → 𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)))
25 eqidd 2728 . . . . . . . . 9 (((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) ∧ 𝑎 = 𝑧) → (𝑔𝑌) = (𝑔𝑌))
26 simpr 484 . . . . . . . . 9 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → 𝑧𝐴)
27 fvexd 6906 . . . . . . . . 9 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → (𝑔𝑌) ∈ V)
28 nfcv 2898 . . . . . . . . . . 11 𝑎𝑓
29 nfmpt1 5250 . . . . . . . . . . 11 𝑎(𝑎𝐴 ↦ (𝑔𝑌))
3028, 29nfeq 2911 . . . . . . . . . 10 𝑎 𝑓 = (𝑎𝐴 ↦ (𝑔𝑌))
31 nfv 1910 . . . . . . . . . 10 𝑎 𝑧𝐴
3230, 31nfan 1895 . . . . . . . . 9 𝑎(𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴)
33 nfcv 2898 . . . . . . . . 9 𝑎𝑧
34 nfcv 2898 . . . . . . . . 9 𝑎(𝑔𝑌)
3524, 25, 26, 27, 32, 33, 34fvmptdf 7005 . . . . . . . 8 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → (𝑓𝑧) = (𝑔𝑌))
3635eqeq1d 2729 . . . . . . 7 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → ((𝑓𝑧) = 𝑏 ↔ (𝑔𝑌) = 𝑏))
3736ralbidva 3170 . . . . . 6 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → (∀𝑧𝐴 (𝑓𝑧) = 𝑏 ↔ ∀𝑧𝐴 (𝑔𝑌) = 𝑏))
3837rexbidv 3173 . . . . 5 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → (∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏 ↔ ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏))
3923, 38anbi12d 630 . . . 4 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → ((𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏) ↔ ((𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏)))
403, 22, 39elabd 3668 . . 3 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)) ∈ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)})
41 cfsetsnfsetfv.f . . 3 𝐹 = {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
4240, 41eleqtrrdi 2839 . 2 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)) ∈ 𝐹)
43 cfsetsnfsetfv.h . 2 𝐻 = (𝑔𝐺 ↦ (𝑎𝐴 ↦ (𝑔𝑌)))
4442, 43fmptd 7118 1 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {cab 2704  wral 3056  wrex 3065  Vcvv 3469  {csn 4624  cmpt 5225  wf 6538  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550
This theorem is referenced by:  cfsetsnfsetf1  46354  cfsetsnfsetfo  46355
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