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Theorem qliftfun 6124
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 (x 𝑋 ↦ ⟨[x]𝑅, A⟩)
qlift.2 ((φ x 𝑋) → A 𝑌)
qlift.3 (φ𝑅 Er 𝑋)
qlift.4 (φ𝑋 V)
qliftfun.4 (x = yA = B)
Assertion
Ref Expression
qliftfun (φ → (Fun 𝐹xy(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 (x 𝑋 ↦ ⟨[x]𝑅, A⟩)
2 qlift.2 . . . 4 ((φ x 𝑋) → A 𝑌)
3 qlift.3 . . . 4 (φ𝑅 Er 𝑋)
4 qlift.4 . . . 4 (φ𝑋 V)
51, 2, 3, 4qliftlem 6120 . . 3 ((φ x 𝑋) → [x]𝑅 (𝑋 / 𝑅))
6 eceq1 6077 . . 3 (x = y → [x]𝑅 = [y]𝑅)
7 qliftfun.4 . . 3 (x = yA = B)
81, 5, 2, 6, 7fliftfun 5379 . 2 (φ → (Fun 𝐹x 𝑋 y 𝑋 ([x]𝑅 = [y]𝑅A = B)))
93adantr 261 . . . . . . . . . . 11 ((φ x𝑅y) → 𝑅 Er 𝑋)
10 simpr 103 . . . . . . . . . . 11 ((φ x𝑅y) → x𝑅y)
119, 10ercl 6053 . . . . . . . . . 10 ((φ x𝑅y) → x 𝑋)
129, 10ercl2 6055 . . . . . . . . . 10 ((φ x𝑅y) → y 𝑋)
1311, 12jca 290 . . . . . . . . 9 ((φ x𝑅y) → (x 𝑋 y 𝑋))
1413ex 108 . . . . . . . 8 (φ → (x𝑅y → (x 𝑋 y 𝑋)))
1514pm4.71rd 374 . . . . . . 7 (φ → (x𝑅y ↔ ((x 𝑋 y 𝑋) x𝑅y)))
163adantr 261 . . . . . . . . 9 ((φ (x 𝑋 y 𝑋)) → 𝑅 Er 𝑋)
17 simprl 483 . . . . . . . . 9 ((φ (x 𝑋 y 𝑋)) → x 𝑋)
1816, 17erth 6086 . . . . . . . 8 ((φ (x 𝑋 y 𝑋)) → (x𝑅y ↔ [x]𝑅 = [y]𝑅))
1918pm5.32da 425 . . . . . . 7 (φ → (((x 𝑋 y 𝑋) x𝑅y) ↔ ((x 𝑋 y 𝑋) [x]𝑅 = [y]𝑅)))
2015, 19bitrd 177 . . . . . 6 (φ → (x𝑅y ↔ ((x 𝑋 y 𝑋) [x]𝑅 = [y]𝑅)))
2120imbi1d 220 . . . . 5 (φ → ((x𝑅yA = B) ↔ (((x 𝑋 y 𝑋) [x]𝑅 = [y]𝑅) → A = B)))
22 impexp 250 . . . . 5 ((((x 𝑋 y 𝑋) [x]𝑅 = [y]𝑅) → A = B) ↔ ((x 𝑋 y 𝑋) → ([x]𝑅 = [y]𝑅A = B)))
2321, 22syl6bb 185 . . . 4 (φ → ((x𝑅yA = B) ↔ ((x 𝑋 y 𝑋) → ([x]𝑅 = [y]𝑅A = B))))
24232albidv 1744 . . 3 (φ → (xy(x𝑅yA = B) ↔ xy((x 𝑋 y 𝑋) → ([x]𝑅 = [y]𝑅A = B))))
25 r2al 2337 . . 3 (x 𝑋 y 𝑋 ([x]𝑅 = [y]𝑅A = B) ↔ xy((x 𝑋 y 𝑋) → ([x]𝑅 = [y]𝑅A = B)))
2624, 25syl6bbr 187 . 2 (φ → (xy(x𝑅yA = B) ↔ x 𝑋 y 𝑋 ([x]𝑅 = [y]𝑅A = B)))
278, 26bitr4d 180 1 (φ → (Fun 𝐹xy(x𝑅yA = B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  cop 3370   class class class wbr 3755  cmpt 3809  ran crn 4289  Fun wfun 4839   Er wer 6039  [cec 6040   / cqs 6041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-er 6042  df-ec 6044  df-qs 6048
This theorem is referenced by:  qliftfund  6125  qliftfuns  6126
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