Theorem xphe 44058

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 6031	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵)
2		rnxpss 6131	. . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
3	1, 2	sstri 3944	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵
4		df-he 44050	. 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵)
5	3, 4	mpbir 231	1 ⊢ (𝐴 × 𝐵) hereditary 𝐵

Syntax hints:   ⊆ wss 3902   × cxp 5623  ran crn 5626   “ cima 5628   hereditary whe 44049

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-he 44050

This theorem is referenced by:  0heALT  44060