Theorem funALTVfun 38683

Description: Our definition of the function predicate df-funALTV 38667 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6501, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)

Assertion

Ref	Expression
funALTVfun	⊢ ( FunALTV 𝐹 ↔ Fun 𝐹)

Proof of Theorem funALTVfun

Step	Hyp	Ref	Expression
1		cnvrefrelcoss2 38521	. . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2		dfcoss3 38398	. . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹)
3	2	sseq1i 3972	. . . 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 38667	. 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7		df-fun 6501	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
8	5, 6, 7	3bitr4i 303	1 ⊢ ( FunALTV 𝐹 ↔ Fun 𝐹)

Syntax hints:   ↔ wb 206   ∧ wa 395   ⊆ wss 3911   I cid 5525  ^◡ccnv 5630   ∘ ccom 5635  Rel wrel 5636  Fun wfun 6493   ≀ 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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-fun 6501  df-coss 38395  df-cnvrefrel 38511  df-funALTV 38667

This theorem is referenced by: (None)