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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 35879 | . . 3 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) | |
| 2 | xpss 5701 | . . 3 ⊢ (V × (V × V)) ⊆ (V × V) | |
| 3 | 1, 2 | sstri 3993 | . 2 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × V) |
| 4 | df-rel 5692 | . 2 ⊢ (Rel (𝐴 ⊗ 𝐵) ↔ (𝐴 ⊗ 𝐵) ⊆ (V × V)) | |
| 5 | 3, 4 | mpbir 231 | 1 ⊢ Rel (𝐴 ⊗ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: Vcvv 3480 ⊆ wss 3951 × cxp 5683 Rel wrel 5690 ⊗ ctxp 35831 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-res 5697 df-txp 35855 |
| This theorem is referenced by: pprodss4v 35885 |
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