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 5926 | . . 3 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) | |
2 | funco 5928 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) | |
3 | 1, 2 | mpan2 707 | . 2 ⊢ (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
4 | df-tpos 7352 | . . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
5 | 4 | funeqi 5909 | . 2 ⊢ (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
6 | 3, 5 | sylibr 224 | 1 ⊢ (Fun 𝐹 → Fun tpos 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∪ cun 3572 ∅c0 3915 {csn 4177 ∪ cuni 4436 ↦ cmpt 4729 ◡ccnv 5113 dom cdm 5114 ∘ ccom 5118 Fun wfun 5882 tpos ctpos 7351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-fun 5890 df-tpos 7352 |
This theorem is referenced by: tposfn2 7374 |
Copyright terms: Public domain | W3C validator |