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| Mirrors > Home > MPE Home > Th. List > Mathboxes > funALTVfun | Structured version Visualization version GIF version | ||
| Description: Our definition of the function predicate df-funALTV 38683 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6563, 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 38538 | . . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I ) | |
| 2 | dfcoss3 38415 | . . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹) | |
| 3 | 2 | sseq1i 4012 | . . . 4 ⊢ ( ≀ 𝐹 ⊆ I ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
| 4 | 1, 3 | bitri 275 | . . 3 ⊢ ( CnvRefRel ≀ 𝐹 ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
| 5 | 4 | anbi2ci 625 | . 2 ⊢ (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) |
| 6 | df-funALTV 38683 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
| 7 | df-fun 6563 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
| 8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ ( FunALTV 𝐹 ↔ Fun 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ⊆ wss 3951 I cid 5577 ◡ccnv 5684 ∘ ccom 5689 Rel wrel 5690 Fun wfun 6555 ≀ ccoss 38182 CnvRefRel wcnvrefrel 38191 FunALTV wfunALTV 38213 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-fun 6563 df-coss 38412 df-cnvrefrel 38528 df-funALTV 38683 |
| This theorem is referenced by: (None) |
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