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Theorem xphe 43770
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 6090 . . 3 ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2 rnxpss 6193 . . 3 ran (𝐴 × 𝐵) ⊆ 𝐵
31, 2sstri 4004 . 2 ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4 df-he 43762 . 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 3962   × cxp 5686  ran crn 5689  cima 5691   hereditary whe 43761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-he 43762
This theorem is referenced by:  0heALT  43772
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