Theorem dffunALTV4 36801

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV4 36827. This is dffun6 6449. 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 36799	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid4 36588	. . 3 ⊢ ( ≀ 𝐹 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
3	2	anbi1i 624	. 2 ⊢ (( ≀ 𝐹 ⊆ I ∧ Rel 𝐹) ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))
4	1, 3	bitri 274	1 ⊢ ( FunALTV 𝐹 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃*wmo 2538   ⊆ wss 3887   class class class wbr 5074   I cid 5488  Rel wrel 5594   ≀ ccoss 36333   FunALTV wfunALTV 36364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-coss 36537  df-cnvrefrel 36643  df-funALTV 36793

This theorem is referenced by: (None)