Theorem xphe 40482

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 5907	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2		rnxpss 5996	. . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
3	1, 2	sstri 3924	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4		df-he 40474	. 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
5	3, 4	mpbir 234	1 ⊢ (𝐴 × 𝐵) hereditary 𝐵

Syntax hints:   ⊆ wss 3881   × cxp 5517  ran crn 5520   “ cima 5522   hereditary whe 40473

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-he 40474

This theorem is referenced by:  0heALT  40484