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Theorem xphe 37557
 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 5436 . . 3 ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2 rnxpss 5525 . . 3 ran (𝐴 × 𝐵) ⊆ 𝐵
31, 2sstri 3592 . 2 ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4 df-he 37549 . 2 ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
53, 4mpbir 221 1 (𝐴 × 𝐵) hereditary 𝐵
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3555   × cxp 5072  ran crn 5075   “ cima 5077   hereditary whe 37548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-he 37549 This theorem is referenced by:  0heALT  37559
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