![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elfunsALTVfunALTV | Structured version Visualization version GIF version |
Description: The element of the class of functions and the function predicate are the same when 𝐹 is a set. (Contributed by Peter Mazsa, 26-Jul-2021.) |
Ref | Expression |
---|---|
elfunsALTVfunALTV | ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossex 37593 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → ≀ 𝐹 ∈ V) | |
2 | elcnvrefrelsrel 37710 | . . . 4 ⊢ ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹)) |
4 | elrelsrel 37661 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Rels ↔ Rel 𝐹)) | |
5 | 3, 4 | anbi12d 630 | . 2 ⊢ (𝐹 ∈ 𝑉 → (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))) |
6 | elfunsALTV 37866 | . 2 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) | |
7 | df-funALTV 37856 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 Vcvv 3473 Rel wrel 5681 ≀ ccoss 37347 Rels crels 37349 CnvRefRels ccnvrefrels 37355 CnvRefRel wcnvrefrel 37356 FunsALTV cfunsALTV 37377 FunALTV wfunALTV 37378 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-coss 37585 df-rels 37659 df-ssr 37672 df-cnvrefs 37699 df-cnvrefrels 37700 df-cnvrefrel 37701 df-funss 37854 df-funsALTV 37855 df-funALTV 37856 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |