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Theorem en2d 6184
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
Hypotheses
Ref Expression
en2d.1  _V
en2d.2  _V
en2d.3  C  _V
en2d.4  D  _V
en2d.5  C  D
Assertion
Ref Expression
en2d  ~~
Distinct variable groups:   ,,   ,,   , C   , D   ,,
Allowed substitution hints:    C()    D()

Proof of Theorem en2d
StepHypRef Expression
1 en2d.1 . 2  _V
2 en2d.2 . 2  _V
3 eqid 2037 . . 3  |->  C  |->  C
4 en2d.3 . . . 4  C  _V
54imp 115 . . 3  C  _V
6 en2d.4 . . . 4  D  _V
76imp 115 . . 3  D  _V
8 en2d.5 . . 3  C  D
93, 5, 7, 8f1od 5645 . 2  |->  C : -1-1-onto->
10 f1oen2g 6171 . 2  _V  _V  |->  C : -1-1-onto->  ~~
111, 2, 9, 10syl3anc 1134 1  ~~
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   _Vcvv 2551   class class class wbr 3755    |-> cmpt 3809   -1-1-onto->wf1o 4844    ~~ cen 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158
This theorem is referenced by:  en2i  6186
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