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Theorem eluniimadm 5367
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 eliun 3658 . 2 (𝐵 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥))
2 funiunfvdm 5365 . . 3 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
32eleq2d 2107 . 2 (𝐹 Fn 𝐴 → (𝐵 𝑥𝐴 (𝐹𝑥) ↔ 𝐵 (𝐹𝐴)))
41, 3syl5rbbr 184 1 (𝐹 Fn 𝐴 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wcel 1393  wrex 2304   cuni 3577   ciun 3654  cima 4311   Fn wfn 4860  cfv 4865
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 3872  ax-pow 3924  ax-pr 3941
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 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-fv 4873
This theorem is referenced by: (None)
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