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Theorem qliftfuns 6126
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)
Assertion
Ref Expression
qliftfuns (φ → (Fun 𝐹yz(y𝑅zy / xA = z / xA)))
Distinct variable groups:   y,z,A   x,y,z,φ   x,𝑅,y,z   y,𝐹,z   x,𝑋,y,z   x,𝑌,y,z
Allowed substitution hints:   A(x)   𝐹(x)

Proof of Theorem qliftfuns
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)
2 nfcv 2175 . . . . 5 y⟨[x]𝑅, A
3 nfcv 2175 . . . . . 6 x[y]𝑅
4 nfcsb1v 2876 . . . . . 6 xy / xA
53, 4nfop 3556 . . . . 5 x⟨[y]𝑅, y / xA
6 eceq1 6077 . . . . . 6 (x = y → [x]𝑅 = [y]𝑅)
7 csbeq1a 2854 . . . . . 6 (x = yA = y / xA)
86, 7opeq12d 3548 . . . . 5 (x = y → ⟨[x]𝑅, A⟩ = ⟨[y]𝑅, y / xA⟩)
92, 5, 8cbvmpt 3842 . . . 4 (x 𝑋 ↦ ⟨[x]𝑅, A⟩) = (y 𝑋 ↦ ⟨[y]𝑅, y / xA⟩)
109rneqi 4505 . . 3 ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩) = ran (y 𝑋 ↦ ⟨[y]𝑅, y / xA⟩)
111, 10eqtri 2057 . 2 𝐹 = ran (y 𝑋 ↦ ⟨[y]𝑅, y / xA⟩)
12 qlift.2 . . . 4 ((φ x 𝑋) → A 𝑌)
1312ralrimiva 2386 . . 3 (φx 𝑋 A 𝑌)
144nfel1 2185 . . . 4 xy / xA 𝑌
157eleq1d 2103 . . . 4 (x = y → (A 𝑌y / xA 𝑌))
1614, 15rspc 2644 . . 3 (y 𝑋 → (x 𝑋 A 𝑌y / xA 𝑌))
1713, 16mpan9 265 . 2 ((φ y 𝑋) → y / xA 𝑌)
18 qlift.3 . 2 (φ𝑅 Er 𝑋)
19 qlift.4 . 2 (φ𝑋 V)
20 csbeq1 2849 . 2 (y = zy / xA = z / xA)
2111, 17, 18, 19, 20qliftfun 6124 1 (φ → (Fun 𝐹yz(y𝑅zy / xA = z / xA)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  csb 2846  cop 3370   class class class wbr 3755  cmpt 3809  ran crn 4289  Fun wfun 4839   Er wer 6039  [cec 6040
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: (None)
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