Theorem funALTVfun 39104

Description: Our definition of the function predicate df-funALTV 39088 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6500, 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 38938	. . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2		dfcoss3 38825	. . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹)
3	2	sseq1i 3950	. . . 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 39088	. 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7		df-fun 6500	. 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 3889   I cid 5525  ^◡ccnv 5630   ∘ ccom 5635  Rel wrel 5636  Fun wfun 6492   ≀ ccoss 38504   CnvRefRel wcnvrefrel 38513   FunALTV wfunALTV 38537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  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 6500  df-coss 38822  df-cnvrefrel 38928  df-funALTV 39088

This theorem is referenced by:  disjqmap2  39147