Theorem xphe 43794

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 6089	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2		rnxpss 6192	. . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
3	1, 2	sstri 3993	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4		df-he 43786	. 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
5	3, 4	mpbir 231	1 ⊢ (𝐴 × 𝐵) hereditary 𝐵

Syntax hints:   ⊆ wss 3951   × cxp 5683  ran crn 5686   “ cima 5688   hereditary whe 43785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-he 43786

This theorem is referenced by:  0heALT  43796