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Theorem fliftfuns 5381
Description: The function 𝐹 is the unique function defined by 𝐹A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
flift.2 ((φ x 𝑋) → A 𝑅)
flift.3 ((φ x 𝑋) → B 𝑆)
Assertion
Ref Expression
fliftfuns (φ → (Fun 𝐹y 𝑋 z 𝑋 (y / xA = z / xAy / xB = z / xB)))
Distinct variable groups:   y,z,A   y,B,z   x,z,y,𝑅   y,𝐹,z   φ,x,y,z   x,𝑋,y,z   x,𝑆,y,z
Allowed substitution hints:   A(x)   B(x)   𝐹(x)

Proof of Theorem fliftfuns
StepHypRef Expression
1 flift.1 . . 3 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
2 nfcv 2175 . . . . 5 yA, B
3 nfcsb1v 2876 . . . . . 6 xy / xA
4 nfcsb1v 2876 . . . . . 6 xy / xB
53, 4nfop 3556 . . . . 5 xy / xA, y / xB
6 csbeq1a 2854 . . . . . 6 (x = yA = y / xA)
7 csbeq1a 2854 . . . . . 6 (x = yB = y / xB)
86, 7opeq12d 3548 . . . . 5 (x = y → ⟨A, B⟩ = ⟨y / xA, y / xB⟩)
92, 5, 8cbvmpt 3842 . . . 4 (x 𝑋 ↦ ⟨A, B⟩) = (y 𝑋 ↦ ⟨y / xA, y / xB⟩)
109rneqi 4505 . . 3 ran (x 𝑋 ↦ ⟨A, B⟩) = ran (y 𝑋 ↦ ⟨y / xA, y / xB⟩)
111, 10eqtri 2057 . 2 𝐹 = ran (y 𝑋 ↦ ⟨y / xA, y / xB⟩)
12 flift.2 . . . 4 ((φ x 𝑋) → A 𝑅)
1312ralrimiva 2386 . . 3 (φx 𝑋 A 𝑅)
143nfel1 2185 . . . 4 xy / xA 𝑅
156eleq1d 2103 . . . 4 (x = y → (A 𝑅y / xA 𝑅))
1614, 15rspc 2644 . . 3 (y 𝑋 → (x 𝑋 A 𝑅y / xA 𝑅))
1713, 16mpan9 265 . 2 ((φ y 𝑋) → y / xA 𝑅)
18 flift.3 . . . 4 ((φ x 𝑋) → B 𝑆)
1918ralrimiva 2386 . . 3 (φx 𝑋 B 𝑆)
204nfel1 2185 . . . 4 xy / xB 𝑆
217eleq1d 2103 . . . 4 (x = y → (B 𝑆y / xB 𝑆))
2220, 21rspc 2644 . . 3 (y 𝑋 → (x 𝑋 B 𝑆y / xB 𝑆))
2319, 22mpan9 265 . 2 ((φ y 𝑋) → y / xB 𝑆)
24 csbeq1 2849 . 2 (y = zy / xA = z / xA)
25 csbeq1 2849 . 2 (y = zy / xB = z / xB)
2611, 17, 23, 24, 25fliftfun 5379 1 (φ → (Fun 𝐹y 𝑋 z 𝑋 (y / xA = z / xAy / xB = z / xB)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  csb 2846  cop 3370  cmpt 3809  ran crn 4289  Fun wfun 4839
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-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
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
This theorem is referenced by: (None)
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