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Theorem fnprOLD 6130
Description: Obsolete version of fnprb 6129 as of 29-Dec-2018. Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) (Revised by NM, 10-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
fnprOLD.1
fnprOLD.2
Assertion
Ref Expression
fnprOLD

Proof of Theorem fnprOLD
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fndm 5685 . . . . 5
2 fvex 5881 . . . . . 6
3 fvex 5881 . . . . . 6
42, 3dmprop 5488 . . . . 5
51, 4syl6eqr 2516 . . . 4
65adantl 466 . . 3
71adantl 466 . . . . . 6
87eleq2d 2527 . . . . 5
9 vex 3112 . . . . . . 7
109elpr 4047 . . . . . 6
11 fnprOLD.1 . . . . . . . . . . 11
1211, 2fvpr1 6114 . . . . . . . . . 10
1312adantr 465 . . . . . . . . 9
1413eqcomd 2465 . . . . . . . 8
15 fveq2 5871 . . . . . . . . 9
16 fveq2 5871 . . . . . . . . 9
1715, 16eqeq12d 2479 . . . . . . . 8
1814, 17syl5ibrcom 222 . . . . . . 7
19 fnprOLD.2 . . . . . . . . . . 11
2019, 3fvpr2 6115 . . . . . . . . . 10
2120adantr 465 . . . . . . . . 9
2221eqcomd 2465 . . . . . . . 8
23 fveq2 5871 . . . . . . . . 9
24 fveq2 5871 . . . . . . . . 9
2523, 24eqeq12d 2479 . . . . . . . 8
2622, 25syl5ibrcom 222 . . . . . . 7
2718, 26jaod 380 . . . . . 6
2810, 27syl5bi 217 . . . . 5
298, 28sylbid 215 . . . 4
3029ralrimiv 2869 . . 3
31 fnfun 5683 . . . 4
3211, 19, 2, 3funpr 5644 . . . 4
33 eqfunfv 5986 . . . 4
3431, 32, 33syl2anr 478 . . 3
356, 30, 34mpbir2and 922 . 2
364a1i 11 . . . 4
37 df-fn 5596 . . . 4
3832, 36, 37sylanbrc 664 . . 3
39 fneq1 5674 . . . 4
4039biimprd 223 . . 3
4138, 40mpan9 469 . 2
4235, 41impbida 832 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807   cvv 3109  {cpr 4031  <.cop 4035  domcdm 5004  Funwfun 5587  Fnwfn 5588  `cfv 5593
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-8 1820  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-pow 4630  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-sbc 3328  df-csb 3435  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-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601
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