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Theorem qliftfun 8842
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 8838 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 8784 . . 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 8756 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑥𝑋)
129, 10ercl2 8758 . . . . . . . . . 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 8796 . . . . . . . 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 1923 . . 3 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
25 r2al 3195 . . 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 1538   = wceq 1540  wcel 2108  wral 3061  cop 4632   class class class wbr 5143  cmpt 5225  ran crn 5686  Fun wfun 6555   Er wer 8742  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-er 8745  df-ec 8747  df-qs 8751
This theorem is referenced by:  qliftfund  8843  qliftfuns  8844
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