Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6386 | . . 3 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) | |
2 | funco 6388 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) | |
3 | 1, 2 | mpan2 687 | . 2 ⊢ (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
4 | df-tpos 7881 | . . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
5 | 4 | funeqi 6369 | . 2 ⊢ (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
6 | 3, 5 | sylibr 235 | 1 ⊢ (Fun 𝐹 → Fun tpos 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3931 ∅c0 4288 {csn 4557 ∪ cuni 4830 ↦ cmpt 5137 ◡ccnv 5547 dom cdm 5548 ∘ ccom 5552 Fun wfun 6342 tpos ctpos 7880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-fun 6350 df-tpos 7881 |
This theorem is referenced by: tposfn2 7903 |
Copyright terms: Public domain | W3C validator |