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Theorem elfunsALTV5 36574
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 36570 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
2 cosselcnvrefrels5 36422 . . . 4 ( ≀ 𝐹 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ ≀ 𝐹 ∈ Rels ))
3 cosselrels 36381 . . . . 5 (𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels )
43biantrud 535 . . . 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 293 . . 3 (𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅)))
65pm5.32ri 579 . 2 (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ 𝐹 ∈ Rels ))
71, 6bitri 278 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 209  wa 399  wo 847   = wceq 1543  wcel 2111  wral 3062  cin 3880  c0 4252  ccnv 5565  ran crn 5567  [cec 8410  ccoss 36100   Rels crels 36102   CnvRefRels ccnvrefrels 36108   FunsALTV cfunsALTV 36130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rmo 3070  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-ec 8414  df-coss 36304  df-rels 36370  df-ssr 36383  df-cnvrefs 36408  df-cnvrefrels 36409  df-funss 36558  df-funsALTV 36559
This theorem is referenced by: (None)
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