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Theorem fvelimab 5229
 Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
Assertion
Ref Expression
fvelimab
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem fvelimab
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2566 . . . 4
21anim2i 324 . . 3
3 ssel2 2940 . . . . . . . 8
4 funfvex 5192 . . . . . . . . 9
54funfni 4999 . . . . . . . 8
63, 5sylan2 270 . . . . . . 7
76anassrs 380 . . . . . 6
8 eleq1 2100 . . . . . 6
97, 8syl5ibcom 144 . . . . 5
109rexlimdva 2433 . . . 4
1110imdistani 419 . . 3
12 eleq1 2100 . . . . . . 7
13 eqeq2 2049 . . . . . . . 8
1413rexbidv 2327 . . . . . . 7
1512, 14bibi12d 224 . . . . . 6
1615imbi2d 219 . . . . 5
17 fnfun 4996 . . . . . . . 8
1817adantr 261 . . . . . . 7
19 fndm 4998 . . . . . . . . 9
2019sseq2d 2973 . . . . . . . 8
2120biimpar 281 . . . . . . 7
22 dfimafn 5222 . . . . . . 7
2318, 21, 22syl2anc 391 . . . . . 6
2423abeq2d 2150 . . . . 5
2516, 24vtoclg 2613 . . . 4
2625impcom 116 . . 3
272, 11, 26pm5.21nd 825 . 2
28 fveq2 5178 . . . 4
2928eqeq1d 2048 . . 3
3029cbvrexv 2534 . 2
3127, 30syl6bb 185 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243   wcel 1393  cab 2026  wrex 2307  cvv 2557   wss 2917   cdm 4345  cima 4348   wfun 4896   wfn 4897  cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910 This theorem is referenced by:  ssimaex  5234  rexima  5394  ralima  5395  f1elima  5412  ovelimab  5651
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