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Theorem dffunsALTV3 37197
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 37195 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2 cosselcnvrefrels3 37051 . . 3 ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑓 ∈ Rels ))
3 cosselrels 37008 . . . 4 (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
43biantrud 533 . . 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 36761 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {crab 3406   class class class wbr 5109  ccoss 36684   Rels crels 36686   CnvRefRels ccnvrefrels 36692   FunsALTV cfunsALTV 36714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-coss 36923  df-rels 36997  df-ssr 37010  df-cnvrefs 37037  df-cnvrefrels 37038  df-funss 37192  df-funsALTV 37193
This theorem is referenced by: (None)
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