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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 35948 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6350, 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 35806 | . . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I ) | |
2 | dfcoss3 35695 | . . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹) | |
3 | 2 | sseq1i 3988 | . . . 4 ⊢ ( ≀ 𝐹 ⊆ I ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
4 | 1, 3 | bitri 277 | . . 3 ⊢ ( CnvRefRel ≀ 𝐹 ↔ (𝐹 ∘ ◡𝐹) ⊆ I ) |
5 | 4 | anbi2ci 626 | . 2 ⊢ (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) |
6 | df-funALTV 35948 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
7 | df-fun 6350 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
8 | 5, 6, 7 | 3bitr4i 305 | 1 ⊢ ( FunALTV 𝐹 ↔ Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ⊆ wss 3929 I cid 5452 ◡ccnv 5547 ∘ ccom 5552 Rel wrel 5553 Fun wfun 6342 ≀ ccoss 35486 CnvRefRel wcnvrefrel 35495 FunALTV wfunALTV 35517 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-fun 6350 df-coss 35692 df-cnvrefrel 35798 df-funALTV 35948 |
This theorem is referenced by: (None) |
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