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Theorem elrnmpt2res 6416
Description: Membership in the range of a restricted operation class abstraction. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypothesis
Ref Expression
rngop.1
Assertion
Ref Expression
elrnmpt2res
Distinct variable groups:   ,   , ,   , ,

Proof of Theorem elrnmpt2res
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2461 . . . . . 6
21anbi1d 704 . . . . 5
32anbi2d 703 . . . 4
432exbidv 1716 . . 3
5 an12 797 . . . . . . . . . 10
6 an12 797 . . . . . . . . . . . 12
7 ancom 450 . . . . . . . . . . . . . 14
8 eleq1 2529 . . . . . . . . . . . . . . . 16
9 df-br 4453 . . . . . . . . . . . . . . . 16
108, 9syl6bbr 263 . . . . . . . . . . . . . . 15
1110anbi2d 703 . . . . . . . . . . . . . 14
127, 11syl5bbr 259 . . . . . . . . . . . . 13
1312anbi2d 703 . . . . . . . . . . . 12
146, 13syl5bb 257 . . . . . . . . . . 11
1514pm5.32i 637 . . . . . . . . . 10
165, 15bitri 249 . . . . . . . . 9
17162exbii 1668 . . . . . . . 8
18 19.42vv 1777 . . . . . . . 8
1917, 18bitr3i 251 . . . . . . 7
2019opabbii 4516 . . . . . 6
21 dfoprab2 6343 . . . . . 6
22 rngop.1 . . . . . . . . 9
23 df-mpt2 6301 . . . . . . . . 9
24 dfoprab2 6343 . . . . . . . . 9
2522, 23, 243eqtri 2490 . . . . . . . 8
2625reseq1i 5274 . . . . . . 7
27 resopab 5325 . . . . . . 7
2826, 27eqtri 2486 . . . . . 6
2920, 21, 283eqtr4ri 2497 . . . . 5
3029rneqi 5234 . . . 4
31 rnoprab 6385 . . . 4
3230, 31eqtri 2486 . . 3
334, 32elab2g 3248 . 2
34 r2ex 2980 . 2
3533, 34syl6bbr 263 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  E.wrex 2808  <.cop 4035   class class class wbr 4452  {copab 4509  rancrn 5005  |`cres 5006  {coprab 6297  e.cmpt2 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-oprab 6300  df-mpt2 6301
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