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Theorem qliftfun 8371
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 8367 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 8316 . . 3 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 qliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
81, 5, 2, 6, 7fliftfun 7054 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
93adantr 481 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10 simpr 485 . . . . . . . . . . 11 ((𝜑𝑥𝑅𝑦) → 𝑥𝑅𝑦)
119, 10ercl 8289 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑥𝑋)
129, 10ercl2 8291 . . . . . . . . . 10 ((𝜑𝑥𝑅𝑦) → 𝑦𝑋)
1311, 12jca 512 . . . . . . . . 9 ((𝜑𝑥𝑅𝑦) → (𝑥𝑋𝑦𝑋))
1413ex 413 . . . . . . . 8 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝑋𝑦𝑋)))
1514pm4.71rd 563 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦)))
163adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑅 Er 𝑋)
17 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
1816, 17erth 8327 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
1918pm5.32da 579 . . . . . . 7 (𝜑 → (((𝑥𝑋𝑦𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2015, 19bitrd 280 . . . . . 6 (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
2120imbi1d 343 . . . . 5 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ (((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22 impexp 451 . . . . 5 ((((𝑥𝑋𝑦𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2321, 22syl6bb 288 . . . 4 (𝜑 → ((𝑥𝑅𝑦𝐴 = 𝐵) ↔ ((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
24232albidv 1915 . . 3 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵))))
25 r2al 3198 . . 3 (∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵) ↔ ∀𝑥𝑦((𝑥𝑋𝑦𝑋) → ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
2624, 25syl6bbr 290 . 2 (𝜑 → (∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵) ↔ ∀𝑥𝑋𝑦𝑋 ([𝑥]𝑅 = [𝑦]𝑅𝐴 = 𝐵)))
278, 26bitr4d 283 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cop 4563   class class class wbr 5057  cmpt 5137  ran crn 5549  Fun wfun 6342   Er wer 8275  [cec 8276   / cqs 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-er 8278  df-ec 8280  df-qs 8284
This theorem is referenced by:  qliftfund  8372  qliftfuns  8373
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