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Theorem eluniimadm 5672
Description: Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
Assertion
Ref Expression
eluniimadm (𝐹 Fn 𝐴 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem eluniimadm
StepHypRef Expression
1 funiunfvdm 5670 . . 3 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
21eleq2d 2210 . 2 (𝐹 Fn 𝐴 → (𝐵 𝑥𝐴 (𝐹𝑥) ↔ 𝐵 (𝐹𝐴)))
3 eliun 3823 . 2 (𝐵 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥))
42, 3bitr3di 194 1 (𝐹 Fn 𝐴 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1481  wrex 2418   cuni 3742   ciun 3819  cima 4548   Fn wfn 5124  cfv 5129
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-fv 5137
This theorem is referenced by:  suplocexprlemell  7543
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