Theorem dffunALTV5 38313

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV5 38339. (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 38310	. 2 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2		cossssid5 38093	. . 3 ⊢ ( ≀ 𝐹 ⊆ I ↔ ∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅))
3	2	anbi1i 622	. 2 ⊢ (( ≀ 𝐹 ⊆ I ∧ Rel 𝐹) ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ Rel 𝐹))
4	1, 3	bitri 274	1 ⊢ ( FunALTV 𝐹 ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ Rel 𝐹))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533  ∀wral 3050   ∩ cin 3943   ⊆ wss 3944  ∅c0 4322   I cid 5575  ^◡ccnv 5677  ran crn 5679  Rel wrel 5683  [cec 8723   ≀ ccoss 37799   FunALTV wfunALTV 37830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rmo 3363  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-coss 38033  df-cnvrefrel 38149  df-funALTV 38304

This theorem is referenced by: (None)