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Theorem enex 6031
Description: The equinumerosity relationship is a set. (Contributed by SF, 23-Feb-2015.)
Assertion
Ref Expression
enex

Proof of Theorem enex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-en 6029 . . 3
2 elrn2 4897 . . . . 5 Fns Image Swap Fns Fns Image Swap Fns
3 df-br 4640 . . . . . . . . 9 Fns Fns
4 vex 2862 . . . . . . . . . 10
54brfns 5833 . . . . . . . . 9 Fns
63, 5bitr3i 242 . . . . . . . 8 Fns
7 elrn2 4897 . . . . . . . . 9 Image Swap Fns Image Swap Fns
8 oteltxp 5782 . . . . . . . . . . . 12 Image Swap Fns Image Swap Fns
9 opelcnv 4893 . . . . . . . . . . . . . 14 Image Swap Image Swap
10 dfcnv2 5100 . . . . . . . . . . . . . . . 16 Swap
1110eqeq2i 2363 . . . . . . . . . . . . . . 15 Swap
12 vex 2862 . . . . . . . . . . . . . . . 16
134, 12brimage 5793 . . . . . . . . . . . . . . 15 Image Swap Swap
14 df-br 4640 . . . . . . . . . . . . . . 15 Image Swap Image Swap
1511, 13, 143bitr2ri 265 . . . . . . . . . . . . . 14 Image Swap
169, 15bitri 240 . . . . . . . . . . . . 13 Image Swap
17 df-br 4640 . . . . . . . . . . . . . 14 Fns Fns
1812brfns 5833 . . . . . . . . . . . . . 14 Fns
1917, 18bitr3i 242 . . . . . . . . . . . . 13 Fns
2016, 19anbi12i 678 . . . . . . . . . . . 12 Image Swap Fns
218, 20bitri 240 . . . . . . . . . . 11 Image Swap Fns
2221exbii 1582 . . . . . . . . . 10 Image Swap Fns
234cnvex 5102 . . . . . . . . . . 11
24 fneq1 5173 . . . . . . . . . . 11
2523, 24ceqsexv 2894 . . . . . . . . . 10
2622, 25bitri 240 . . . . . . . . 9 Image Swap Fns
277, 26bitri 240 . . . . . . . 8 Image Swap Fns
286, 27anbi12i 678 . . . . . . 7 Fns Image Swap Fns
29 oteltxp 5782 . . . . . . 7 Fns Image Swap Fns Fns Image Swap Fns
30 dff1o4 5294 . . . . . . 7
3128, 29, 303bitr4i 268 . . . . . 6 Fns Image Swap Fns
3231exbii 1582 . . . . 5 Fns Image Swap Fns
332, 32bitri 240 . . . 4 Fns Image Swap Fns
3433opabbi2i 4866 . . 3 Fns Image Swap Fns
351, 34eqtr4i 2376 . 2 Fns Image Swap Fns
36 fnsex 5832 . . . 4 Fns
37 swapex 4742 . . . . . . . 8 Swap
3837imageex 5801 . . . . . . 7 Image Swap
3938cnvex 5102 . . . . . 6 Image Swap
4039, 36txpex 5785 . . . . 5 Image Swap Fns
4140rnex 5107 . . . 4 Image Swap Fns
4236, 41txpex 5785 . . 3 Fns Image Swap Fns
4342rnex 5107 . 2 Fns Image Swap Fns
4435, 43eqeltri 2423 1
Colors of variables: wff setvar class
Syntax hints:   wa 358  wex 1541   wceq 1642   wcel 1710  cvv 2859  cop 4561  copab 4622   class class class wbr 4639   Swap cswap 4718  cima 4722  ccnv 4771   crn 4773   wfn 4776  wf1o 4780   ctxp 5735  Imagecimage 5753   Fns cfns 5761   cen 6028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-en 6029
This theorem is referenced by:  ener  6039  ncsex  6111  ncex  6117  ovmuc  6130  mucex  6133  ovcelem1  6171  ceex  6174
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