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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 6071 | . . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵) | |
2 | rnxpss 6172 | . . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
3 | 1, 2 | sstri 3992 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵 |
4 | df-he 42524 | . 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 3949 × cxp 5675 ran crn 5678 “ cima 5680 hereditary whe 42523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-he 42524 |
This theorem is referenced by: 0heALT 42534 |
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