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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 5943 | . . 3 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ ran (𝐴 × 𝐵) | |
2 | rnxpss 6032 | . . 3 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
3 | 1, 2 | sstri 3979 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵 |
4 | df-he 40125 | . 2 ⊢ ((𝐴 × 𝐵) hereditary 𝐵 ↔ ((𝐴 × 𝐵) “ 𝐵) ⊆ 𝐵) | |
5 | 3, 4 | mpbir 233 | 1 ⊢ (𝐴 × 𝐵) hereditary 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3939 × cxp 5556 ran crn 5559 “ cima 5561 hereditary whe 40124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-he 40125 |
This theorem is referenced by: 0heALT 40135 |
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