Theorem funALTVfun 38979

Description: Our definition of the function predicate df-funALTV 38963 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6494, 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 38812	. . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2		dfcoss3 38699	. . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹)
3	2	sseq1i 3962	. . . 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 38963	. 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7		df-fun 6494	. 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 3901   I cid 5518  ^◡ccnv 5623   ∘ ccom 5628  Rel wrel 5629  Fun wfun 6486   ≀ ccoss 38386   CnvRefRel wcnvrefrel 38395   FunALTV wfunALTV 38417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-fun 6494  df-coss 38696  df-cnvrefrel 38802  df-funALTV 38963

This theorem is referenced by: (None)