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Mirrors > Home > MPE Home > Th. List > Mathboxes > txprel | Structured version Visualization version GIF version |
Description: A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
txprel | ⊢ Rel (𝐴 ⊗ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txpss3v 34395 | . . 3 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) | |
2 | xpss 5647 | . . 3 ⊢ (V × (V × V)) ⊆ (V × V) | |
3 | 1, 2 | sstri 3951 | . 2 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × V) |
4 | df-rel 5638 | . 2 ⊢ (Rel (𝐴 ⊗ 𝐵) ↔ (𝐴 ⊗ 𝐵) ⊆ (V × V)) | |
5 | 3, 4 | mpbir 230 | 1 ⊢ Rel (𝐴 ⊗ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3443 ⊆ wss 3908 × cxp 5629 Rel wrel 5636 ⊗ ctxp 34347 |
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-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
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-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-res 5643 df-txp 34371 |
This theorem is referenced by: pprodss4v 34401 |
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