Theorem funALTVfun 38697

Description: Our definition of the function predicate df-funALTV 38681 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6516, 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 38535	. . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2		dfcoss3 38412	. . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹)
3	2	sseq1i 3978	. . . 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 38681	. 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7		df-fun 6516	. 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 3917   I cid 5535  ^◡ccnv 5640   ∘ ccom 5645  Rel wrel 5646  Fun wfun 6508   ≀ ccoss 38176   CnvRefRel wcnvrefrel 38185   FunALTV wfunALTV 38207

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-fun 6516  df-coss 38409  df-cnvrefrel 38525  df-funALTV 38681

This theorem is referenced by: (None)