Theorem dffunALTV4 39850

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV4 39876. This is dffun6 6558. 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 39848	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid4 39635	. . 3 ⊢ ( ≀ 𝐹 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
3	2	anbi1i 636	. 2 ⊢ (( ≀ 𝐹 ⊆ I ∧ Rel 𝐹) ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))
4	1, 3	bitri 278	1 ⊢ ( FunALTV 𝐹 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃*wmo 2565   ⊆ wss 3900   class class class wbr 5106   I cid 5549  Rel wrel 5660   ≀ ccoss 39258   FunALTV wfunALTV 39291

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-coss 39576  df-cnvrefrel 39682  df-funALTV 39842

This theorem is used by: (None)