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Theorem xpheOLD 36423
Description: Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) Obsolete version of xphe 36422 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
xpheOLD  |-  ( A  X.  B ) hereditary  B

Proof of Theorem xpheOLD
StepHypRef Expression
1 df-res 4868 . . . 4  |-  ( ( A  X.  B )  |`  B )  =  ( ( A  X.  B
)  i^i  ( B  X.  _V ) )
2 inxp 4989 . . . 4  |-  ( ( A  X.  B )  i^i  ( B  X.  _V ) )  =  ( ( A  i^i  B
)  X.  ( B  i^i  _V ) )
3 inv1 3773 . . . . 5  |-  ( B  i^i  _V )  =  B
43xpeq2i 4877 . . . 4  |-  ( ( A  i^i  B )  X.  ( B  i^i  _V ) )  =  ( ( A  i^i  B
)  X.  B )
51, 2, 43eqtri 2488 . . 3  |-  ( ( A  X.  B )  |`  B )  =  ( ( A  i^i  B
)  X.  B )
6 inss2 3665 . . . 4  |-  ( A  i^i  B )  C_  B
7 xpss1 4965 . . . 4  |-  ( ( A  i^i  B ) 
C_  B  ->  (
( A  i^i  B
)  X.  B ) 
C_  ( B  X.  B ) )
86, 7ax-mp 5 . . 3  |-  ( ( A  i^i  B )  X.  B )  C_  ( B  X.  B
)
95, 8eqsstri 3474 . 2  |-  ( ( A  X.  B )  |`  B )  C_  ( B  X.  B )
10 dfhe2 36415 . 2  |-  ( ( A  X.  B ) hereditary  B 
<->  ( ( A  X.  B )  |`  B ) 
C_  ( B  X.  B ) )
119, 10mpbir 214 1  |-  ( A  X.  B ) hereditary  B
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3057    i^i cin 3415    C_ wss 3416    X. cxp 4854    |` cres 4858   hereditary whe 36413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-xp 4862  df-rel 4863  df-cnv 4864  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-he 36414
This theorem is referenced by: (None)
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