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Theorem rabfmpunirn 27815
 Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
Hypotheses
Ref Expression
rabfmpunirn.1
rabfmpunirn.2
rabfmpunirn.3
Assertion
Ref Expression
rabfmpunirn
Distinct variable groups:   ,,   ,,   ,,   ,   ,
Allowed substitution hints:   (,)   ()   ()

Proof of Theorem rabfmpunirn
StepHypRef Expression
1 rabfmpunirn.1 . . . 4
2 df-rab 2760 . . . . 5
32mpteq2i 4475 . . . 4
41, 3eqtri 2429 . . 3
5 rabfmpunirn.2 . . . 4
65zfausab 4540 . . 3
7 eleq1 2472 . . . 4
8 rabfmpunirn.3 . . . 4
97, 8anbi12d 709 . . 3
104, 6, 9abfmpunirn 27814 . 2
11 elex 3065 . . . . 5
1211adantr 463 . . . 4
1312rexlimivw 2890 . . 3
1413pm4.71ri 631 . 2
1510, 14bitr4i 252 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1403   wcel 1840  cab 2385  wrex 2752  crab 2755  cvv 3056  cuni 4188   cmpt 4450   crn 4941 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531 This theorem is referenced by: (None)
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