Theorem si3ex 5807

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

Step	Hyp	Ref	Expression
1		df-si3 5759	. 2 ⊢ SI3 A = (( SI 1st ⊗ ( SI (1st ∘ 2nd ) ⊗ SI (2nd ∘ 2nd ))) “ ℘1A)
2		1stex 4740	. . . . 5 ⊢ 1st ∈ V
3	2	siex 4754	. . . 4 ⊢ SI 1st ∈ V
4		2ndex 5113	. . . . . . 7 ⊢ 2nd ∈ V
5	2, 4	coex 4751	. . . . . 6 ⊢ (1st ∘ 2nd ) ∈ V
6	5	siex 4754	. . . . 5 ⊢ SI (1st ∘ 2nd ) ∈ V
7	4, 4	coex 4751	. . . . . 6 ⊢ (2nd ∘ 2nd ) ∈ V
8	7	siex 4754	. . . . 5 ⊢ SI (2nd ∘ 2nd ) ∈ V
9	6, 8	txpex 5786	. . . 4 ⊢ ( SI (1st ∘ 2nd ) ⊗ SI (2nd ∘ 2nd )) ∈ V
10	3, 9	txpex 5786	. . 3 ⊢ ( SI 1st ⊗ ( SI (1st ∘ 2nd ) ⊗ SI (2nd ∘ 2nd ))) ∈ V
11		si3ex.1	. . . 4 ⊢ A ∈ V
12	11	pw1ex 4304	. . 3 ⊢ ℘1A ∈ V
13	10, 12	imaex 4748	. 2 ⊢ (( SI 1st ⊗ ( SI (1st ∘ 2nd ) ⊗ SI (2nd ∘ 2nd ))) “ ℘1A) ∈ V
14	1, 13	eqeltri 2423	1 ⊢ SI3 A ∈ V

Syntax hints:   ∈ wcel 1710  Vcvv 2860  ℘₁cpw1 4136  1^st c1st 4718   SI csi 4721   ∘ ccom 4722   “ cima 4723  2^nd c2nd 4784   ⊗ ctxp 5736   SI₃ csi3 5758

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-co 4727  df-ima 4728  df-si 4729  df-cnv 4786  df-2nd 4798  df-txp 5737  df-si3 5759

This theorem is referenced by:  composeex  5821  addcfnex  5825  funsex  5829  crossex  5851  domfnex  5871  ranfnex  5872  transex  5911  antisymex  5913  connexex  5914  foundex  5915  extex  5916  symex  5917  mucex  6134  ovcelem1  6172  ceex  6175