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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 37194 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6502, 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 37049 | . . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I ) | |
2 | dfcoss3 36926 | . . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹) | |
3 | 2 | sseq1i 3976 | . . . 4 ⊢ ( ≀ 𝐹 ⊆ I ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
4 | 1, 3 | bitri 275 | . . 3 ⊢ ( CnvRefRel ≀ 𝐹 ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
5 | 4 | anbi2ci 626 | . 2 ⊢ (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) |
6 | df-funALTV 37194 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
7 | df-fun 6502 | . 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 205 ∧ wa 397 ⊆ wss 3914 I cid 5534 ◡ccnv 5636 ∘ ccom 5641 Rel wrel 5642 Fun wfun 6494 ≀ ccoss 36684 CnvRefRel wcnvrefrel 36693 FunALTV wfunALTV 36715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-fun 6502 df-coss 36923 df-cnvrefrel 37039 df-funALTV 37194 |
This theorem is referenced by: (None) |
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