Theorem dffunALTV4 38808

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV4 38834. This is dffun6 6497. 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 38806	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid4 38592	. . 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 1539  ∃*wmo 2535   ⊆ wss 3898   class class class wbr 5093   I cid 5513  Rel wrel 5624   ≀ ccoss 38242   FunALTV wfunALTV 38273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-coss 38533  df-cnvrefrel 38639  df-funALTV 38800

This theorem is referenced by: (None)