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| Mirrors > Home > MPE Home > Th. List > relfull | Structured version Visualization version GIF version | ||
| Description: The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
| Ref | Expression |
|---|---|
| relfull | ⊢ Rel (𝐶 Full 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fullfunc 17953 | . 2 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷) | |
| 2 | relfunc 17907 | . 2 ⊢ Rel (𝐶 Func 𝐷) | |
| 3 | relss 5791 | . 2 ⊢ ((𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷) → (Rel (𝐶 Func 𝐷) → Rel (𝐶 Full 𝐷))) | |
| 4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel (𝐶 Full 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 Rel wrel 5690 (class class class)co 7431 Func cfunc 17899 Full cful 17949 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-func 17903 df-full 17951 |
| This theorem is referenced by: fullpropd 17967 cofull 17981 thincciso 49102 thincciso2 49104 |
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