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Theorem ins4ex 5799
 Description: Ins4 preserves sethood. (Contributed by SF, 12-Feb-2015.)
Hypothesis
Ref Expression
insex.1 A V
Assertion
Ref Expression
ins4ex Ins4 A V

Proof of Theorem ins4ex
StepHypRef Expression
1 df-ins4 5756 . 2 Ins4 A = ((1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) “ A)
2 1stex 4739 . . . . 5 1st V
3 2ndex 5112 . . . . . . 7 2nd V
42, 3coex 4750 . . . . . 6 (1st 2nd ) V
54, 3coex 4750 . . . . . 6 ((1st 2nd ) 2nd ) V
64, 5txpex 5785 . . . . 5 ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )) V
72, 6txpex 5785 . . . 4 (1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) V
87cnvex 5102 . . 3 (1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) V
9 insex.1 . . 3 A V
108, 9imaex 4747 . 2 ((1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) “ A) V
111, 10eqeltri 2423 1 Ins4 A V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710  Vcvv 2859  1st c1st 4717   ∘ ccom 4721   “ cima 4722  ◡ccnv 4771  2nd c2nd 4783   ⊗ ctxp 5735   Ins4 cins4 5755 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-cnv 4785  df-2nd 4797  df-txp 5736  df-ins4 5756 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  ovmuc  6130  mucex  6133  ovcelem1  6171  ceex  6174
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