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Theorem dffunsALTV3 38667
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 38665 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2 cosselcnvrefrels3 38521 . . 3 ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels ))
3 cosselrels 38478 . . . 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 38233 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  {crab 3433   class class class wbr 5148  ccoss 38162   Rels crels 38164   CnvRefRels ccnvrefrels 38170   FunsALTV cfunsALTV 38192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-coss 38393  df-rels 38467  df-ssr 38480  df-cnvrefs 38507  df-cnvrefrels 38508  df-funss 38662  df-funsALTV 38663
This theorem is referenced by: (None)
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