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Theorem eluniima 6390
Description: Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)
Assertion
Ref Expression
eluniima (Fun 𝐹 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem eluniima
StepHypRef Expression
1 eliun 4454 . 2 (𝐵 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥))
2 funiunfv 6388 . . 3 (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
32eleq2d 2672 . 2 (Fun 𝐹 → (𝐵 𝑥𝐴 (𝐹𝑥) ↔ 𝐵 (𝐹𝐴)))
41, 3syl5rbbr 273 1 (Fun 𝐹 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wcel 1976  wrex 2896   cuni 4366   ciun 4449  cima 5031  Fun wfun 5784  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  elunirnALT  6392  alephfp  8791  acsficl2d  16945  elhf  31257
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