MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qliftfund Structured version   Visualization version   GIF version

Theorem qliftfund 8103
Description: The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2 ((𝜑𝑥𝑋) → 𝐴𝑌)
qlift.3 (𝜑𝑅 Er 𝑋)
qlift.4 (𝜑𝑋 ∈ V)
qliftfun.4 (𝑥 = 𝑦𝐴 = 𝐵)
qliftfund.6 ((𝜑𝑥𝑅𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
qliftfund (𝜑 → Fun 𝐹)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦,𝜑   𝑥,𝑅,𝑦   𝑦,𝐹   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐹(𝑥)

Proof of Theorem qliftfund
StepHypRef Expression
1 qliftfund.6 . . . 4 ((𝜑𝑥𝑅𝑦) → 𝐴 = 𝐵)
21ex 403 . . 3 (𝜑 → (𝑥𝑅𝑦𝐴 = 𝐵))
32alrimivv 2027 . 2 (𝜑 → ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵))
4 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
5 qlift.2 . . 3 ((𝜑𝑥𝑋) → 𝐴𝑌)
6 qlift.3 . . 3 (𝜑𝑅 Er 𝑋)
7 qlift.4 . . 3 (𝜑𝑋 ∈ V)
8 qliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
94, 5, 6, 7, 8qliftfun 8102 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
103, 9mpbird 249 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1654   = wceq 1656  wcel 2164  Vcvv 3414  cop 4405   class class class wbr 4875  cmpt 4954  ran crn 5347  Fun wfun 6121   Er wer 8011  [cec 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-er 8014  df-ec 8016  df-qs 8020
This theorem is referenced by:  orbstafun  18101  frgpupf  18546
  Copyright terms: Public domain W3C validator