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Theorem cfsetsnfsetf 44239
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 486 . . . . . 6 ((𝐴𝑉𝑌𝐴) → 𝐴𝑉)
21adantr 484 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → 𝐴𝑉)
32mptexd 7049 . . . 4 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)) ∈ V)
4 vex 3419 . . . . . . . . . . 11 𝑔 ∈ V
5 feq1 6535 . . . . . . . . . . 11 (𝑥 = 𝑔 → (𝑥:{𝑌}⟶𝐵𝑔:{𝑌}⟶𝐵))
6 cfsetsnfsetfv.g . . . . . . . . . . 11 𝐺 = {𝑥𝑥:{𝑌}⟶𝐵}
74, 5, 6elab2 3598 . . . . . . . . . 10 (𝑔𝐺𝑔:{𝑌}⟶𝐵)
87biimpi 219 . . . . . . . . 9 (𝑔𝐺𝑔:{𝑌}⟶𝐵)
98adantl 485 . . . . . . . 8 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → 𝑔:{𝑌}⟶𝐵)
10 snidg 4584 . . . . . . . . . 10 (𝑌𝐴𝑌 ∈ {𝑌})
1110adantl 485 . . . . . . . . 9 ((𝐴𝑉𝑌𝐴) → 𝑌 ∈ {𝑌})
1211adantr 484 . . . . . . . 8 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → 𝑌 ∈ {𝑌})
139, 12ffvelrnd 6914 . . . . . . 7 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑔𝑌) ∈ 𝐵)
1413adantr 484 . . . . . 6 ((((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) ∧ 𝑎𝐴) → (𝑔𝑌) ∈ 𝐵)
1514fmpttd 6941 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵)
16 eqeq2 2750 . . . . . . . 8 (𝑏 = (𝑔𝑌) → ((𝑔𝑌) = 𝑏 ↔ (𝑔𝑌) = (𝑔𝑌)))
1716ralbidv 3119 . . . . . . 7 (𝑏 = (𝑔𝑌) → (∀𝑧𝐴 (𝑔𝑌) = 𝑏 ↔ ∀𝑧𝐴 (𝑔𝑌) = (𝑔𝑌)))
1817adantl 485 . . . . . 6 ((((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) ∧ 𝑏 = (𝑔𝑌)) → (∀𝑧𝐴 (𝑔𝑌) = 𝑏 ↔ ∀𝑧𝐴 (𝑔𝑌) = (𝑔𝑌)))
19 eqidd 2739 . . . . . . 7 ((((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) ∧ 𝑧𝐴) → (𝑔𝑌) = (𝑔𝑌))
2019ralrimiva 3106 . . . . . 6 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → ∀𝑧𝐴 (𝑔𝑌) = (𝑔𝑌))
2113, 18, 20rspcedvd 3547 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏)
2215, 21jca 515 . . . 4 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → ((𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏))
23 feq1 6535 . . . . 5 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → (𝑓:𝐴𝐵 ↔ (𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵))
24 simpl 486 . . . . . . . . 9 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → 𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)))
25 eqidd 2739 . . . . . . . . 9 (((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) ∧ 𝑎 = 𝑧) → (𝑔𝑌) = (𝑔𝑌))
26 simpr 488 . . . . . . . . 9 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → 𝑧𝐴)
27 fvexd 6741 . . . . . . . . 9 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → (𝑔𝑌) ∈ V)
28 nfcv 2905 . . . . . . . . . . 11 𝑎𝑓
29 nfmpt1 5162 . . . . . . . . . . 11 𝑎(𝑎𝐴 ↦ (𝑔𝑌))
3028, 29nfeq 2918 . . . . . . . . . 10 𝑎 𝑓 = (𝑎𝐴 ↦ (𝑔𝑌))
31 nfv 1922 . . . . . . . . . 10 𝑎 𝑧𝐴
3230, 31nfan 1907 . . . . . . . . 9 𝑎(𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴)
33 nfcv 2905 . . . . . . . . 9 𝑎𝑧
34 nfcv 2905 . . . . . . . . 9 𝑎(𝑔𝑌)
3524, 25, 26, 27, 32, 33, 34fvmptdf 6833 . . . . . . . 8 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → (𝑓𝑧) = (𝑔𝑌))
3635eqeq1d 2740 . . . . . . 7 ((𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) ∧ 𝑧𝐴) → ((𝑓𝑧) = 𝑏 ↔ (𝑔𝑌) = 𝑏))
3736ralbidva 3118 . . . . . 6 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → (∀𝑧𝐴 (𝑓𝑧) = 𝑏 ↔ ∀𝑧𝐴 (𝑔𝑌) = 𝑏))
3837rexbidv 3223 . . . . 5 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → (∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏 ↔ ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏))
3923, 38anbi12d 634 . . . 4 (𝑓 = (𝑎𝐴 ↦ (𝑔𝑌)) → ((𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏) ↔ ((𝑎𝐴 ↦ (𝑔𝑌)):𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑔𝑌) = 𝑏)))
403, 22, 39elabd 3597 . . 3 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)) ∈ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)})
41 cfsetsnfsetfv.f . . 3 𝐹 = {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
4240, 41eleqtrrdi 2850 . 2 (((𝐴𝑉𝑌𝐴) ∧ 𝑔𝐺) → (𝑎𝐴 ↦ (𝑔𝑌)) ∈ 𝐹)
43 cfsetsnfsetfv.h . 2 𝐻 = (𝑔𝐺 ↦ (𝑎𝐴 ↦ (𝑔𝑌)))
4442, 43fmptd 6940 1 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  {cab 2715  wral 3062  wrex 3063  Vcvv 3415  {csn 4550  cmpt 5144  wf 6385  cfv 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5188  ax-sep 5201  ax-nul 5208  ax-pr 5331
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-f1 6394  df-fo 6395  df-f1o 6396  df-fv 6397
This theorem is referenced by:  cfsetsnfsetf1  44240  cfsetsnfsetfo  44241
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