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Theorem elfunsALTV5 38677
Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
elfunsALTV5 (𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ 𝐹 ∈ Rels ))
Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem elfunsALTV5
StepHypRef Expression
1 elfunsALTV 38673 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
2 cosselcnvrefrels5 38522 . . . 4 ( ≀ 𝐹 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ ≀ 𝐹 ∈ Rels ))
3 cosselrels 38477 . . . . 5 (𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels )
43biantrud 531 . . . 4 (𝐹 ∈ Rels → (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ ≀ 𝐹 ∈ Rels )))
52, 4bitr4id 290 . . 3 (𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅)))
65pm5.32ri 575 . 2 (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ 𝐹 ∈ Rels ))
71, 6bitri 275 1 (𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ 𝐹 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wral 3058  cin 3961  c0 4338  ccnv 5687  ran crn 5689  [cec 8741  ccoss 38161   Rels crels 38163   CnvRefRels ccnvrefrels 38169   FunsALTV cfunsALTV 38191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745  df-coss 38392  df-rels 38466  df-ssr 38479  df-cnvrefs 38506  df-cnvrefrels 38507  df-funss 38661  df-funsALTV 38662
This theorem is referenced by: (None)
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