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Theorem xpheOLD 36229
Description: Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) Obsolete version of xphe 36228 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 4858 . . . 4  |-  ( ( A  X.  B )  |`  B )  =  ( ( A  X.  B
)  i^i  ( B  X.  _V ) )
2 inxp 4979 . . . 4  |-  ( ( A  X.  B )  i^i  ( B  X.  _V ) )  =  ( ( A  i^i  B
)  X.  ( B  i^i  _V ) )
3 inv1 3786 . . . . 5  |-  ( B  i^i  _V )  =  B
43xpeq2i 4867 . . . 4  |-  ( ( A  i^i  B )  X.  ( B  i^i  _V ) )  =  ( ( A  i^i  B
)  X.  B )
51, 2, 43eqtri 2453 . . 3  |-  ( ( A  X.  B )  |`  B )  =  ( ( A  i^i  B
)  X.  B )
6 inss2 3680 . . . 4  |-  ( A  i^i  B )  C_  B
7 xpss1 4955 . . . 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 3491 . 2  |-  ( ( A  X.  B )  |`  B )  C_  ( B  X.  B )
10 dfhe2 36221 . 2  |-  ( ( A  X.  B ) hereditary  B 
<->  ( ( A  X.  B )  |`  B ) 
C_  ( B  X.  B ) )
119, 10mpbir 212 1  |-  ( A  X.  B ) hereditary  B
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3078    i^i cin 3432    C_ wss 3433    X. cxp 4844    |` cres 4848   hereditary whe 36219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-xp 4852  df-rel 4853  df-cnv 4854  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-he 36220
This theorem is referenced by: (None)
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