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

Theorem funiunfvdmf 5658
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5657 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
Hypothesis
Ref Expression
funiunfvf.1 𝑥𝐹
Assertion
Ref Expression
funiunfvdmf (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem funiunfvdmf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 funiunfvf.1 . . . 4 𝑥𝐹
2 nfcv 2279 . . . 4 𝑥𝑧
31, 2nffv 5424 . . 3 𝑥(𝐹𝑧)
4 nfcv 2279 . . 3 𝑧(𝐹𝑥)
5 fveq2 5414 . . 3 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
63, 4, 5cbviun 3845 . 2 𝑧𝐴 (𝐹𝑧) = 𝑥𝐴 (𝐹𝑥)
7 funiunfvdm 5657 . 2 (𝐹 Fn 𝐴 𝑧𝐴 (𝐹𝑧) = (𝐹𝐴))
86, 7syl5eqr 2184 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wnfc 2266   cuni 3731   ciun 3808  cima 4537   Fn wfn 5113  cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator