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Theorem si3ex 5806
 Description: SI3 preserves sethood. (Contributed by SF, 12-Feb-2015.)
Hypothesis
Ref Expression
si3ex.1 A V
Assertion
Ref Expression
si3ex SI3 A V

Proof of Theorem si3ex
StepHypRef Expression
1 df-si3 5758 . 2 SI3 A = (( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) “ 1A)
2 1stex 4739 . . . . 5 1st V
32siex 4753 . . . 4 SI 1st V
4 2ndex 5112 . . . . . . 7 2nd V
52, 4coex 4750 . . . . . 6 (1st 2nd ) V
65siex 4753 . . . . 5 SI (1st 2nd ) V
74, 4coex 4750 . . . . . 6 (2nd 2nd ) V
87siex 4753 . . . . 5 SI (2nd 2nd ) V
96, 8txpex 5785 . . . 4 ( SI (1st 2nd ) ⊗ SI (2nd 2nd )) V
103, 9txpex 5785 . . 3 ( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) V
11 si3ex.1 . . . 4 A V
1211pw1ex 4303 . . 3 1A V
1310, 12imaex 4747 . 2 (( SI 1st ⊗ ( SI (1st 2nd ) ⊗ SI (2nd 2nd ))) “ 1A) V
141, 13eqeltri 2423 1 SI3 A V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710  Vcvv 2859  ℘1cpw1 4135  1st c1st 4717   SI csi 4720   ∘ ccom 4721   “ cima 4722  2nd c2nd 4783   ⊗ ctxp 5735   SI3 csi3 5757 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-co 4726  df-ima 4727  df-si 4728  df-cnv 4785  df-2nd 4797  df-txp 5736  df-si3 5758 This theorem is referenced by:  composeex  5820  addcfnex  5824  funsex  5828  crossex  5850  domfnex  5870  ranfnex  5871  transex  5910  antisymex  5912  connexex  5913  foundex  5914  extex  5915  symex  5916  mucex  6133  ovcelem1  6171  ceex  6174
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