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Theorem fnprOLD 6115
 Description: Obsolete version of fnprb 6114 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 5670 . . . . 5
2 fvex 5866 . . . . . 6
3 fvex 5866 . . . . . 6
42, 3dmprop 5473 . . . . 5
51, 4syl6eqr 2502 . . . 4
71adantl 466 . . . . . 6
87eleq2d 2513 . . . . 5
9 vex 3098 . . . . . . 7
109elpr 4032 . . . . . 6
11 fnprOLD.1 . . . . . . . . . . 11
1211, 2fvpr1 6099 . . . . . . . . . 10
1312adantr 465 . . . . . . . . 9
1413eqcomd 2451 . . . . . . . 8
15 fveq2 5856 . . . . . . . . 9
16 fveq2 5856 . . . . . . . . 9
1715, 16eqeq12d 2465 . . . . . . . 8
1814, 17syl5ibrcom 222 . . . . . . 7
19 fnprOLD.2 . . . . . . . . . . 11
2019, 3fvpr2 6100 . . . . . . . . . 10
2120adantr 465 . . . . . . . . 9
2221eqcomd 2451 . . . . . . . 8
23 fveq2 5856 . . . . . . . . 9
24 fveq2 5856 . . . . . . . . 9
2523, 24eqeq12d 2465 . . . . . . . 8
2622, 25syl5ibrcom 222 . . . . . . 7
2718, 26jaod 380 . . . . . 6
2810, 27syl5bi 217 . . . . 5
298, 28sylbid 215 . . . 4
3029ralrimiv 2855 . . 3
31 fnfun 5668 . . . 4
3211, 19, 2, 3funpr 5629 . . . 4
33 eqfunfv 5971 . . . 4
3431, 32, 33syl2anr 478 . . 3
356, 30, 34mpbir2and 922 . 2
364a1i 11 . . . 4
37 df-fn 5581 . . . 4
3832, 36, 37sylanbrc 664 . . 3
39 fneq1 5659 . . . 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 1383   wcel 1804   wne 2638  wral 2793  cvv 3095  cpr 4016  cop 4020   cdm 4989   wfun 5572   wfn 5573  cfv 5578 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586 This theorem is referenced by: (None)
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