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Theorem elunirnALT 6876
Description: Alternate proof of elunirn 6875. It is shorter but requires ax-pow 5157 (through eluniima 6874, funiunfv 6872, ndmfv 6568). (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 5816 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
21unieqi 4754 . . 3 (𝐹 “ dom 𝐹) = ran 𝐹
32eleq2i 2874 . 2 (𝐴 (𝐹 “ dom 𝐹) ↔ 𝐴 ran 𝐹)
4 eluniima 6874 . 2 (Fun 𝐹 → (𝐴 (𝐹 “ dom 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
53, 4syl5bbr 286 1 (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wcel 2081  wrex 3106   cuni 4745  dom cdm 5443  ran crn 5444  cima 5446  Fun wfun 6219  cfv 6225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-fv 6233
This theorem is referenced by: (None)
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