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Theorem qliftfun 7719
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)
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)

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 (𝜑𝑋 ∈ V)
51, 2, 3, 4qliftlem 7715 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 7669 . . 3 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 qliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
81, 5, 2, 6, 7fliftfun 6462 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
93adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10 simpr 476 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑥𝑅𝑦)
119, 10ercl 7640 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑥𝑋)
129, 10ercl2 7642 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑦𝑋)
1311, 12jca 553 . . . . . . . . 9 ((𝜑𝑥𝑅𝑦) → (𝑥𝑋𝑦𝑋))
1413ex 449 . . . . . . . 8 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝑋𝑦𝑋)))
1514pm4.71rd 665 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦)))
163adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑅 Er 𝑋)
17 simprl 790 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
1816, 17erth 7678 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
1918pm5.32da 671 . . . . . . 7 (𝜑 → (((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2015, 19bitrd 267 . . . . . 6 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2120imbi1d 330 . . . . 5 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ (((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22 impexp 461 . . . . 5 ((((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2321, 22syl6bb 275 . . . 4 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
24232albidv 1838 . . 3 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
25 r2al 2923 . . 3 (∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2624, 25syl6bbr 277 . 2 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
278, 26bitr4d 270 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cop 4131   class class class wbr 4583  cmpt 4643  ran crn 5039  Fun wfun 5798   Er wer 7626  [cec 7627   / cqs 7628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-er 7629  df-ec 7631  df-qs 7635
This theorem is referenced by:  qliftfund  7720  qliftfuns  7721
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