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Theorem selsubf 25990
 Description: A way of selecting a subset of functions so that their values belong to . (Contributed by FL, 14-Jan-2014.)
Hypotheses
Ref Expression
selsubf.1
selsubf.2
Assertion
Ref Expression
selsubf

Proof of Theorem selsubf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 in32 3381 . . 3
2 xpindi 4819 . . . . . 6
32pweqi 3629 . . . . 5
4 pwin 4297 . . . . 5
53, 4eqtr2i 2304 . . . 4
65ineq1i 3366 . . 3
71, 6eqtri 2303 . 2
8 selsubf.1 . . . 4
9 selsubf.2 . . . 4
108, 9mapval2 6797 . . 3
1110ineq1i 3366 . 2
128inex1 4155 . . 3
1312, 9mapval2 6797 . 2
147, 11, 133eqtr4i 2313 1
 Colors of variables: wff set class Syntax hints:   wceq 1623   wcel 1684  cab 2269  cvv 2788   cin 3151  cpw 3625   cxp 4687   wfn 5250  (class class class)co 5858   cmap 6772 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774
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