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Theorem elunirnALT 6001
 Description: Membership in the union of the range of a function, proved directly. Unlike elunirn 5999, it doesn't appeal to ndmfv 5756 (via funiunfv 5996). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elunirnALT
Distinct variable groups:   ,   ,

Proof of Theorem elunirnALT
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni 4019 . 2
2 funfn 5483 . . . . . . . 8
3 fvelrnb 5775 . . . . . . . 8
42, 3sylbi 189 . . . . . . 7
54anbi2d 686 . . . . . 6
6 r19.42v 2863 . . . . . 6
75, 6syl6bbr 256 . . . . 5
8 eleq2 2498 . . . . . . 7
98biimparc 475 . . . . . 6
109reximi 2814 . . . . 5
117, 10syl6bi 221 . . . 4
1211exlimdv 1647 . . 3
13 fvelrn 5867 . . . . . . 7
1413a1d 24 . . . . . 6
1514ancld 538 . . . . 5
16 fvex 5743 . . . . . 6
17 eleq2 2498 . . . . . . 7
18 eleq1 2497 . . . . . . 7
1917, 18anbi12d 693 . . . . . 6
2016, 19spcev 3044 . . . . 5
2115, 20syl6 32 . . . 4
2221rexlimdva 2831 . . 3
2312, 22impbid 185 . 2
241, 23syl5bb 250 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  wrex 2707  cuni 4016   cdm 4879   crn 4880   wfun 5449   wfn 5450  cfv 5455 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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