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Theorem pw1exg 4302
Description: The unit power class preserves sethood. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
pw1exg (A V1A V)

Proof of Theorem pw1exg
StepHypRef Expression
1 dfpw12 4301 . 2 1A = ( SIk (A ×k A) “k V)
2 xpkexg 4288 . . . . 5 ((A V A V) → (A ×k A) V)
32anidms 626 . . . 4 (A V → (A ×k A) V)
4 sikexg 4296 . . . 4 ((A ×k A) V → SIk (A ×k A) V)
53, 4syl 15 . . 3 (A VSIk (A ×k A) V)
6 vvex 4109 . . 3 V V
7 imakexg 4299 . . 3 (( SIk (A ×k A) V V V) → ( SIk (A ×k A) “k V) V)
85, 6, 7sylancl 643 . 2 (A V → ( SIk (A ×k A) “k V) V)
91, 8syl5eqel 2437 1 (A V1A V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   wcel 1710  Vcvv 2859  1cpw1 4135   ×k cxpk 4174  k cimak 4179   SIk csik 4181
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-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-si 4083  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192
This theorem is referenced by:  pw1ex  4303  pw1exb  4326  pwexg  4328  addcexg  4393  ncfintfin  4495  imaexg  4746  coexg  4749  siexg  4752  qsexg  5982  pw1eltc  6162  ncpw1pwneg  6201  ltcpw1pwg  6202  tcncg  6224  canncb  6332
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