Theorem elfunsALTV5 39036

Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 5-Sep-2021.)

Assertion

Ref	Expression
elfunsALTV5	⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ 𝐹 ∈ Rels ))

Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem elfunsALTV5

Step	Hyp	Ref	Expression
1		elfunsALTV 39032	. 2 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
2		cosselcnvrefrels5 38876	. . . 4 ⊢ ( ≀ 𝐹 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ ≀ 𝐹 ∈ Rels ))
3		cosselrels 38830	. . . . 5 ⊢ (𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels )
4	3	biantrud 531	. . . 4 ⊢ (𝐹 ∈ Rels → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ ≀ 𝐹 ∈ Rels )))
5	2, 4	bitr4id 290	. . 3 ⊢ (𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅)))
6	5	pm5.32ri 575	. 2 ⊢ (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ 𝐹 ∈ Rels ))
7	1, 6	bitri 275	1 ⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ 𝐹 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902  ∅c0 4287  ^◡ccnv 5631  ran crn 5633  [cec 8643   ≀ ccoss 38438   Rels crels 38440   CnvRefRels ccnvrefrels 38446   FunsALTV cfunsALTV 38470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rmo 3352  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647  df-rels 38695  df-coss 38756  df-ssr 38833  df-cnvrefs 38860  df-cnvrefrels 38861  df-funss 39020  df-funsALTV 39021

This theorem is referenced by: (None)