Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cfsetsnfsetfv Structured version   Visualization version   GIF version

Theorem cfsetsnfsetfv 46498
Description: The function value of the mapping of the class of singleton functions into the class of constant functions. (Contributed by AV, 13-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
cfsetsnfsetfv ((𝐴𝑉𝑋𝐺) → (𝐻𝑋) = (𝑎𝐴 ↦ (𝑋𝑌)))
Distinct variable groups:   𝐴,𝑎,𝑔   𝑔,𝐺   𝑔,𝑉   𝑋,𝑎,𝑔   𝑔,𝑌
Allowed substitution hints:   𝐴(𝑥,𝑧,𝑓,𝑏)   𝐵(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝐹(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝐺(𝑥,𝑧,𝑓,𝑎,𝑏)   𝐻(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑧,𝑓,𝑎,𝑏)   𝑋(𝑥,𝑧,𝑓,𝑏)   𝑌(𝑥,𝑧,𝑓,𝑎,𝑏)

Proof of Theorem cfsetsnfsetfv
StepHypRef Expression
1 cfsetsnfsetfv.h . . 3 𝐻 = (𝑔𝐺 ↦ (𝑎𝐴 ↦ (𝑔𝑌)))
21a1i 11 . 2 ((𝐴𝑉𝑋𝐺) → 𝐻 = (𝑔𝐺 ↦ (𝑎𝐴 ↦ (𝑔𝑌))))
3 fveq1 6889 . . . . 5 (𝑔 = 𝑋 → (𝑔𝑌) = (𝑋𝑌))
43adantr 479 . . . 4 ((𝑔 = 𝑋𝑎𝐴) → (𝑔𝑌) = (𝑋𝑌))
54mpteq2dva 5244 . . 3 (𝑔 = 𝑋 → (𝑎𝐴 ↦ (𝑔𝑌)) = (𝑎𝐴 ↦ (𝑋𝑌)))
65adantl 480 . 2 (((𝐴𝑉𝑋𝐺) ∧ 𝑔 = 𝑋) → (𝑎𝐴 ↦ (𝑔𝑌)) = (𝑎𝐴 ↦ (𝑋𝑌)))
7 simpr 483 . 2 ((𝐴𝑉𝑋𝐺) → 𝑋𝐺)
8 simpl 481 . . 3 ((𝐴𝑉𝑋𝐺) → 𝐴𝑉)
98mptexd 7230 . 2 ((𝐴𝑉𝑋𝐺) → (𝑎𝐴 ↦ (𝑋𝑌)) ∈ V)
102, 6, 7, 9fvmptd 7005 1 ((𝐴𝑉𝑋𝐺) → (𝐻𝑋) = (𝑎𝐴 ↦ (𝑋𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2702  wral 3051  wrex 3060  Vcvv 3463  {csn 4625  cmpt 5227  wf 6539  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  cfsetsnfsetf1  46500  cfsetsnfsetfo  46501
  Copyright terms: Public domain W3C validator