Theorem xphe 38075

Description: Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)

Assertion

Ref	Expression
xphe	|- ( A X. B ) hereditary B

Proof of Theorem xphe

Step	Hyp	Ref	Expression
1		imassrn 5477	. . 3 |- ( ( A X. B ) " B ) C_ ran ( A X. B )
2		rnxpss 5566	. . 3 |- ran ( A X. B ) C_ B
3	1, 2	sstri 3612	. 2 |- ( ( A X. B ) " B ) C_ B
4		df-he 38067	. 2 |- ( ( A X. B ) hereditary B <-> ( ( A X. B ) " B ) C_ B )
5	3, 4	mpbir 221	1 |- ( A X. B ) hereditary B

Syntax hints:   C_ wss 3574   X. cxp 5112   ran crn 5115   "cima 5117   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  0heALT  38077