Theorem dffunALTV5 38656

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV5 38682. (Contributed by Peter Mazsa, 5-Sep-2021.)

Assertion

Ref	Expression
dffunALTV5	⊢ ( FunALTV 𝐹 ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ Rel 𝐹))

Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem dffunALTV5

Step	Hyp	Ref	Expression
1		dffunALTV2 38653	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid5 38435	. . 3 ⊢ ( ≀ 𝐹 ⊆ I ↔ ∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅))
3	2	anbi1i 624	. 2 ⊢ (( ≀ 𝐹 ⊆ I ∧ Rel 𝐹) ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ Rel 𝐹))
4	1, 3	bitri 275	1 ⊢ ( FunALTV 𝐹 ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ Rel 𝐹))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∀wral 3044   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292   I cid 5525  ^◡ccnv 5630  ran crn 5632  Rel wrel 5636  [cec 8646   ≀ ccoss 38142   FunALTV wfunALTV 38173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rmo 3351  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-coss 38375  df-cnvrefrel 38491  df-funALTV 38647

This theorem is referenced by: (None)