Theorem dffunALTV4 38713

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV4 38739. This is dffun6 6549. For the 𝑋 axis and the 𝑌 axis you can convert the right side to (∀𝑥₁∃*𝑦₁𝑥₁𝐹𝑦₁ ∧ Rel 𝐹). (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
dffunALTV4	⊢ ( FunALTV 𝐹 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))

Distinct variable group:   𝑢,𝐹,𝑥

Proof of Theorem dffunALTV4

Step	Hyp	Ref	Expression
1		dffunALTV2 38711	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid4 38493	. . 3 ⊢ ( ≀ 𝐹 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
3	2	anbi1i 624	. 2 ⊢ (( ≀ 𝐹 ⊆ I ∧ Rel 𝐹) ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))
4	1, 3	bitri 275	1 ⊢ ( FunALTV 𝐹 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃*wmo 2538   ⊆ wss 3931   class class class wbr 5124   I cid 5552  Rel wrel 5664   ≀ ccoss 38204   FunALTV wfunALTV 38235

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-coss 38434  df-cnvrefrel 38550  df-funALTV 38705

This theorem is referenced by: (None)