![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > xphe | Structured version Visualization version GIF version |
Description: Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.) |
Ref | Expression |
---|---|
xphe | ⊢ (𝐴 × 𝐵) hereditary 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 6090 | . . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵) | |
2 | rnxpss 6193 | . . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
3 | 1, 2 | sstri 4004 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵 |
4 | df-he 43762 | . 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵) | |
5 | 3, 4 | mpbir 231 | 1 ⊢ (𝐴 × 𝐵) hereditary 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3962 × cxp 5686 ran crn 5689 “ cima 5691 hereditary whe 43761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-he 43762 |
This theorem is referenced by: 0heALT 43772 |
Copyright terms: Public domain | W3C validator |