Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xphe Structured version   Visualization version   GIF version

Theorem xphe 40414
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 5918 . . 3 ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2 rnxpss 6007 . . 3 ran (𝐴 × 𝐵) ⊆ 𝐵
31, 2sstri 3951 . 2 ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4 df-he 40406 . 2 ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
53, 4mpbir 234 1 (𝐴 × 𝐵) hereditary 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3908   × cxp 5530  ran crn 5533  cima 5535   hereditary whe 40405
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-he 40406
This theorem is referenced by:  0heALT  40416
  Copyright terms: Public domain W3C validator