ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funfvdm2f GIF version

Theorem funfvdm2f 5238
Description: The value of a function. Version of funfvdm2 5237 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
Hypotheses
Ref Expression
funfvdm2f.1 𝑦𝐴
funfvdm2f.2 𝑦𝐹
Assertion
Ref Expression
funfvdm2f ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Proof of Theorem funfvdm2f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 funfvdm2 5237 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfvdm2f.1 . . . . 5 𝑦𝐴
3 funfvdm2f.2 . . . . 5 𝑦𝐹
4 nfcv 2178 . . . . 5 𝑦𝑤
52, 3, 4nfbr 3808 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1421 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 3768 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvab 2160 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 3590 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9syl6eq 2088 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  {cab 2026  wnfc 2165   cuni 3580   class class class wbr 3764  dom cdm 4345  Fun wfun 4896  cfv 4902
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator