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Theorem bnj142OLD 31998
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 5981.
Assertion
Ref Expression
bnj142OLD

Proof of Theorem bnj142OLD
StepHypRef Expression
1 fnresdm 5602 . . . 4
2 fnfun 5590 . . . . 5
3 funressn 5978 . . . . 5
42, 3syl 16 . . . 4
51, 4eqsstr3d 3473 . . 3
65sseld 3437 . 2
7 elsni 3984 . 2
86, 7syl6 33 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1370  e.wcel 1757  C_wss 3410  {csn 3959  <.cop 3965  |`cres 4924  Funwfun 5494  Fnwfn 5495  `cfv 5500
This theorem is referenced by:  bnj145OLD  31999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4495  ax-nul 4503  ax-pr 4613
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3054  df-sbc 3269  df-dif 3413  df-un 3415  df-in 3417  df-ss 3424  df-nul 3720  df-if 3874  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4174  df-br 4375  df-opab 4433  df-id 4718  df-xp 4928  df-rel 4929  df-cnv 4930  df-co 4931  df-dm 4932  df-rn 4933  df-res 4934  df-ima 4935  df-iota 5463  df-fun 5502  df-fn 5503  df-f 5504  df-f1 5505  df-fo 5506  df-f1o 5507  df-fv 5508
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