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Theorem dffunsALTV3 38687
Description: Alternate definition of the class of functions. For the 𝑋 axis and the 𝑌 axis you can convert the right side to {𝑓 ∈ Rels ∣ ∀ x1 y1 y2 (( x1 𝑓 y1 x1 𝑓 y2 ) → y1 = y2 )}. (Contributed by Peter Mazsa, 30-Aug-2021.)
Assertion
Ref Expression
dffunsALTV3 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦)}
Distinct variable group:   𝑢,𝑓,𝑥,𝑦

Proof of Theorem dffunsALTV3
StepHypRef Expression
1 dffunsALTV 38685 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2 cosselcnvrefrels3 38541 . . 3 ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels ))
3 cosselrels 38498 . . . 4 (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
43biantrud 531 . . 3 (𝑓 ∈ Rels → (∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦) ↔ (∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels )))
52, 4bitr4id 290 . 2 (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦)))
61, 5rabimbieq 38253 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537   = wceq 1539  wcel 2107  {crab 3435   class class class wbr 5142  ccoss 38183   Rels crels 38185   CnvRefRels ccnvrefrels 38191   FunsALTV cfunsALTV 38213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-coss 38413  df-rels 38487  df-ssr 38500  df-cnvrefs 38527  df-cnvrefrels 38528  df-funss 38682  df-funsALTV 38683
This theorem is referenced by: (None)
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