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Theorem eluniimadm 5296
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) ↔ x A B (𝐹x)))
Distinct variable groups:   x,A   x,B   x,𝐹

Proof of Theorem eluniimadm
StepHypRef Expression
1 eliun 3613 . 2 (B x A (𝐹x) ↔ x A B (𝐹x))
2 funiunfvdm 5294 . . 3 (𝐹 Fn A x A (𝐹x) = (𝐹A))
32eleq2d 2089 . 2 (𝐹 Fn A → (B x A (𝐹x) ↔ B (𝐹A)))
41, 3syl5rbbr 184 1 (𝐹 Fn A → (B (𝐹A) ↔ x A B (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1375  wrex 2283   cuni 3532   ciun 3609  cima 4241   Fn wfn 4791  cfv 4796
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 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804
This theorem is referenced by: (None)
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