Theorem xphe 41359

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

Step	Hyp	Ref	Expression
1		imassrn 5982	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2		rnxpss 6077	. . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
3	1, 2	sstri 3931	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4		df-he 41351	. 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
5	3, 4	mpbir 230	1 ⊢ (𝐴 × 𝐵) hereditary 𝐵

Syntax hints:   ⊆ wss 3888   × cxp 5589  ran crn 5592   “ cima 5594   hereditary whe 41350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-br 5077  df-opab 5139  df-xp 5597  df-rel 5598  df-cnv 5599  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-he 41351

This theorem is referenced by:  0heALT  41361