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Theorem xphe 43743
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 × 𝐵) hereditary 𝐵

Proof of Theorem xphe
StepHypRef Expression
1 imassrn 6100 . . 3 ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2 rnxpss 6203 . . 3 ran (𝐴 × 𝐵) ⊆ 𝐵
31, 2sstri 4018 . 2 ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4 df-he 43735 . 2 ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
53, 4mpbir 231 1 (𝐴 × 𝐵) hereditary 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3976   × cxp 5698  ran crn 5701  cima 5703   hereditary whe 43734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-he 43735
This theorem is referenced by:  0heALT  43745
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