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Mathbox for Peter Mazsa |
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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 38664 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6565, 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 38519 | . . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I ) | |
2 | dfcoss3 38396 | . . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹) | |
3 | 2 | sseq1i 4024 | . . . 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 38664 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
7 | df-fun 6565 | . 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 3963 I cid 5582 ◡ccnv 5688 ∘ ccom 5693 Rel wrel 5694 Fun wfun 6557 ≀ ccoss 38162 CnvRefRel wcnvrefrel 38171 FunALTV wfunALTV 38193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-fun 6565 df-coss 38393 df-cnvrefrel 38509 df-funALTV 38664 |
This theorem is referenced by: (None) |
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