Theorem xphe 42208

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 6044	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2		rnxpss 6144	. . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
3	1, 2	sstri 3971	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4		df-he 42200	. 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
5	3, 4	mpbir 230	1 ⊢ (𝐴 × 𝐵) hereditary 𝐵

Syntax hints:   ⊆ wss 3928   × cxp 5651  ran crn 5654   “ cima 5656   hereditary whe 42199

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-br 5126  df-opab 5188  df-xp 5659  df-rel 5660  df-cnv 5661  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-he 42200

This theorem is referenced by:  0heALT  42210