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Theorem qliftfun 8484
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 8480 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 8429 . . 3 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 qliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
81, 5, 2, 6, 7fliftfun 7121 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
93adantr 484 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10 simpr 488 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑥𝑅𝑦)
119, 10ercl 8402 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑥𝑋)
129, 10ercl2 8404 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑦𝑋)
1311, 12jca 515 . . . . . . . . 9 ((𝜑𝑥𝑅𝑦) → (𝑥𝑋𝑦𝑋))
1413ex 416 . . . . . . . 8 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝑋𝑦𝑋)))
1514pm4.71rd 566 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦)))
163adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑅 Er 𝑋)
17 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
1816, 17erth 8440 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
1918pm5.32da 582 . . . . . . 7 (𝜑 → (((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2015, 19bitrd 282 . . . . . 6 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2120imbi1d 345 . . . . 5 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ (((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22 impexp 454 . . . . 5 ((((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2321, 22bitrdi 290 . . . 4 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
24232albidv 1931 . . 3 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
25 r2al 3122 . . 3 (∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2624, 25bitr4di 292 . 2 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
278, 26bitr4d 285 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wcel 2110  wral 3061  cop 4547   class class class wbr 5053  cmpt 5135  ran crn 5552  Fun wfun 6374   Er wer 8388  [cec 8389   / cqs 8390
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 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
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 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-er 8391  df-ec 8393  df-qs 8397
This theorem is referenced by:  qliftfund  8485  qliftfuns  8486
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