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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 6029 | . . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵) | |
2 | rnxpss 6129 | . . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
3 | 1, 2 | sstri 3956 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵 |
4 | df-he 42167 | . 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵) | |
5 | 3, 4 | mpbir 230 | 1 ⊢ (𝐴 × 𝐵) hereditary 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3913 × cxp 5636 ran crn 5639 “ cima 5641 hereditary whe 42166 |
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 5261 ax-nul 5268 ax-pr 5389 |
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 3406 df-v 3448 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-he 42167 |
This theorem is referenced by: 0heALT 42177 |
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