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Theorem xphe 42175
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 6029 . . 3 ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2 rnxpss 6129 . . 3 ran (𝐴 × 𝐵) ⊆ 𝐵
31, 2sstri 3956 . 2 ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4 df-he 42167 . 2 ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
53, 4mpbir 230 1 (𝐴 × 𝐵) hereditary 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3913   × cxp 5636  ran crn 5639  cima 5641   hereditary whe 42166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-he 42167
This theorem is referenced by:  0heALT  42177
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