![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > funALTVfun | Structured version Visualization version GIF version |
Description: Our definition of the function predicate df-funALTV 36075 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6326, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.) |
Ref | Expression |
---|---|
funALTVfun | ⊢ ( FunALTV 𝐹 ↔ Fun 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvrefrelcoss2 35933 | . . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I ) | |
2 | dfcoss3 35822 | . . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹) | |
3 | 2 | sseq1i 3943 | . . . 4 ⊢ ( ≀ 𝐹 ⊆ I ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
4 | 1, 3 | bitri 278 | . . 3 ⊢ ( CnvRefRel ≀ 𝐹 ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
5 | 4 | anbi2ci 627 | . 2 ⊢ (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) |
6 | df-funALTV 36075 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
7 | df-fun 6326 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
8 | 5, 6, 7 | 3bitr4i 306 | 1 ⊢ ( FunALTV 𝐹 ↔ Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ⊆ wss 3881 I cid 5424 ◡ccnv 5518 ∘ ccom 5523 Rel wrel 5524 Fun wfun 6318 ≀ ccoss 35613 CnvRefRel wcnvrefrel 35622 FunALTV wfunALTV 35644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-fun 6326 df-coss 35819 df-cnvrefrel 35925 df-funALTV 36075 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |