Theorem funALTVfun 39460

Description: Our definition of the function predicate df-funALTV 39444 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6538, 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 39294	. . . 4 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2		dfcoss3 39181	. . . . 5 ⊢ ≀ 𝐹 = (𝐹 ∘ ◡𝐹)
3	2	sseq1i 3964	. . . 4 ⊢ ( ≀ 𝐹 ⊆ I ↔ (𝐹 ∘ ◡𝐹) ⊆ I )
4	1, 3	bitri 278	. . 3 ⊢ ( CnvRefRel ≀ 𝐹 ↔ (𝐹 ∘ ◡𝐹) ⊆ I )
5	4	anbi2ci 636	. 2 ⊢ (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
6		df-funALTV 39444	. 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7		df-fun 6538	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
8	5, 6, 7	3bitr4i 306	1 ⊢ ( FunALTV 𝐹 ↔ Fun 𝐹)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ⊆ wss 3904   I cid 5554  ^◡ccnv 5659   ∘ ccom 5664  Rel wrel 5665  Fun wfun 6530   ≀ ccoss 38860   CnvRefRel wcnvrefrel 38869   FunALTV wfunALTV 38893

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-fun 6538  df-coss 39178  df-cnvrefrel 39284  df-funALTV 39444

This theorem is used by:  disjqmap2  39503