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Theorem elunirnALT 7003
 Description: Alternate proof of elunirn 7002. It is shorter but requires ax-pow 5234 (through eluniima 7001, funiunfv 6999, ndmfv 6688). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elunirnALT (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem elunirnALT
StepHypRef Expression
1 imadmrn 5911 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
21unieqi 4811 . . 3 (𝐹 “ dom 𝐹) = ran 𝐹
32eleq2i 2843 . 2 (𝐴 (𝐹 “ dom 𝐹) ↔ 𝐴 ran 𝐹)
4 eluniima 7001 . 2 (Fun 𝐹 → (𝐴 (𝐹 “ dom 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
53, 4bitr3id 288 1 (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2111  ∃wrex 3071  ∪ cuni 4798  dom cdm 5524  ran crn 5525   “ cima 5527  Fun wfun 6329  ‘cfv 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-fv 6343 This theorem is referenced by: (None)
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