Theorem dffunALTV3 39707

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV3 39733. Reproduction of dffun2 6548. For the 𝑋 axis and the 𝑌 axis you can convert the right side to (∀ x1 ∀ y1 ∀ y2 (( x1 𝑓 y1 ∧ x1 𝑓 y2 ) → y1 = y2 ) ∧ Rel 𝐹). (Contributed by NM, 29-Dec-1996.)

Assertion

Ref	Expression
dffunALTV3	⊢ ( FunALTV 𝐹 ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))

Distinct variable group:   𝑢,𝐹,𝑥,𝑦

Proof of Theorem dffunALTV3

Step	Hyp	Ref	Expression
1		dffunALTV2 39706	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid3 39492	. . 3 ⊢ ( ≀ 𝐹 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦))
3	2	anbi1i 636	. 2 ⊢ (( ≀ 𝐹 ⊆ I ∧ Rel 𝐹) ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))
4	1, 3	bitri 278	1 ⊢ ( FunALTV 𝐹 ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   ⊆ wss 3895   class class class wbr 5102   I cid 5544  Rel wrel 5655   ≀ ccoss 39116   FunALTV wfunALTV 39149

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2099  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-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-coss 39434  df-cnvrefrel 39540  df-funALTV 39700

This theorem is used by: (None)