Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvmpts GIF version

Theorem fvmpts 5193
 Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fvmpts.1 𝐹 = (x 𝐶B)
Assertion
Ref Expression
fvmpts ((A 𝐶 A / xB 𝑉) → (𝐹A) = A / xB)
Distinct variable group:   x,𝐶
Allowed substitution hints:   A(x)   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmpts
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2849 . 2 (y = Ay / xB = A / xB)
2 fvmpts.1 . . 3 𝐹 = (x 𝐶B)
3 nfcv 2175 . . . 4 yB
4 nfcsb1v 2876 . . . 4 xy / xB
5 csbeq1a 2854 . . . 4 (x = yB = y / xB)
63, 4, 5cbvmpt 3842 . . 3 (x 𝐶B) = (y 𝐶y / xB)
72, 6eqtri 2057 . 2 𝐹 = (y 𝐶y / xB)
81, 7fvmptg 5191 1 ((A 𝐶 A / xB 𝑉) → (𝐹A) = A / xB)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ⦋csb 2846   ↦ cmpt 3809  ‘cfv 4845 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-bndl 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-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-iota 4810  df-fun 4847  df-fv 4853 This theorem is referenced by:  fvmptd  5196
 Copyright terms: Public domain W3C validator