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Mirrors > Home > MPE Home > Th. List > tposfun | Structured version Visualization version GIF version |
Description: The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposfun | ⊢ (Fun 𝐹 → Fun tpos 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6395 | . . 3 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) | |
2 | funco 6397 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
4 | df-tpos 7894 | . . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
5 | 4 | funeqi 6378 | . 2 ⊢ (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
6 | 3, 5 | sylibr 236 | 1 ⊢ (Fun 𝐹 → Fun tpos 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3936 ∅c0 4293 {csn 4569 ∪ cuni 4840 ↦ cmpt 5148 ◡ccnv 5556 dom cdm 5557 ∘ ccom 5561 Fun wfun 6351 tpos ctpos 7893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-fun 6359 df-tpos 7894 |
This theorem is referenced by: tposfn2 7916 |
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