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

Theorem qliftfun 8841
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.) (Revised by AV, 3-Aug-2024.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2 ((𝜑𝑥𝑋) → 𝐴𝑌)
qlift.3 (𝜑𝑅 Er 𝑋)
qlift.4 (𝜑𝑋𝑉)
qliftfun.4 (𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
qliftfun (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
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)   𝑉(𝑥,𝑦)

Proof of Theorem qliftfun
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
3 qlift.3 . . . 4 (𝜑𝑅 Er 𝑋)
4 qlift.4 . . . 4 (𝜑𝑋𝑉)
51, 2, 3, 4qliftlem 8837 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 8783 . . 3 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 qliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
81, 5, 2, 6, 7fliftfun 7332 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
93adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10 simpr 484 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑥𝑅𝑦)
119, 10ercl 8755 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑥𝑋)
129, 10ercl2 8757 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑦𝑋)
1311, 12jca 511 . . . . . . . . 9 ((𝜑𝑥𝑅𝑦) → (𝑥𝑋𝑦𝑋))
1413ex 412 . . . . . . . 8 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝑋𝑦𝑋)))
1514pm4.71rd 562 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦)))
163adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑅 Er 𝑋)
17 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
1816, 17erth 8795 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
1918pm5.32da 579 . . . . . . 7 (𝜑 → (((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2015, 19bitrd 279 . . . . . 6 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2120imbi1d 341 . . . . 5 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ (((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22 impexp 450 . . . . 5 ((((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2321, 22bitrdi 287 . . . 4 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
24232albidv 1921 . . 3 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
25 r2al 3193 . . 3 (∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2624, 25bitr4di 289 . 2 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
278, 26bitr4d 282 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wral 3059  cop 4637   class class class wbr 5148  cmpt 5231  ran crn 5690  Fun wfun 6557   Er wer 8741  [cec 8742   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-er 8744  df-ec 8746  df-qs 8750
This theorem is referenced by:  qliftfund  8842  qliftfuns  8843
  Copyright terms: Public domain W3C validator