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

Theorem qliftfuns 8001
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)
Assertion
Ref Expression
qliftfuns (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧,𝜑   𝑥,𝑅,𝑦,𝑧   𝑦,𝐹,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem qliftfuns
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 nfcv 2902 . . . . 5 𝑦⟨[𝑥]𝑅, 𝐴
3 nfcv 2902 . . . . . 6 𝑥[𝑦]𝑅
4 nfcsb1v 3690 . . . . . 6 𝑥𝑦 / 𝑥𝐴
53, 4nfop 4569 . . . . 5 𝑥⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴
6 eceq1 7949 . . . . . 6 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 csbeq1a 3683 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
86, 7opeq12d 4561 . . . . 5 (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
92, 5, 8cbvmpt 4901 . . . 4 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
109rneqi 5507 . . 3 ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
111, 10eqtri 2782 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
12 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
1312ralrimiva 3104 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
144nfel1 2917 . . . 4 𝑥𝑦 / 𝑥𝐴𝑌
157eleq1d 2824 . . . 4 (𝑥 = 𝑦 → (𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1614, 15rspc 3443 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1713, 16mpan9 487 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑌)
18 qlift.3 . 2 (𝜑𝑅 Er 𝑋)
19 qlift.4 . 2 (𝜑𝑋 ∈ V)
20 csbeq1 3677 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2111, 17, 18, 19, 20qliftfun 7999 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  csb 3674  cop 4327   class class class wbr 4804  cmpt 4881  ran crn 5267  Fun wfun 6043   Er wer 7908  [cec 7909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-er 7911  df-ec 7913  df-qs 7917
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator