Theorem dffunALTV4 39526

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV4 39552. This is dffun6 6544. 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 39524	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid4 39311	. . 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 1568  ∃*wmo 2562   ⊆ wss 3899   class class class wbr 5103   I cid 5549  Rel wrel 5660   ≀ ccoss 38934   FunALTV wfunALTV 38967

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  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 39252  df-cnvrefrel 39358  df-funALTV 39518

This theorem is used by: (None)