Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dffunsALTV2 Structured version   Visualization version   GIF version

Theorem dffunsALTV2 38589
Description: Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
Assertion
Ref Expression
dffunsALTV2 FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }

Proof of Theorem dffunsALTV2
StepHypRef Expression
1 dffunsALTV 38588 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2 cosselcnvrefrels2 38443 . . 3 ( ≀ 𝑓 ∈ CnvRefRels ↔ ( ≀ 𝑓 ⊆ I ∧ ≀ 𝑓 ∈ Rels ))
3 cosselrels 38401 . . . 4 (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
43biantrud 531 . . 3 (𝑓 ∈ Rels → ( ≀ 𝑓 ⊆ I ↔ ( ≀ 𝑓 ⊆ I ∧ ≀ 𝑓 ∈ Rels )))
52, 4bitr4id 290 . 2 (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ≀ 𝑓 ⊆ I ))
61, 5rabimbieq 38156 1 FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2103  {crab 3438  wss 3970   I cid 5596  ccoss 38084   Rels crels 38086   CnvRefRels ccnvrefrels 38092   FunsALTV cfunsALTV 38114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-coss 38316  df-rels 38390  df-ssr 38403  df-cnvrefs 38430  df-cnvrefrels 38431  df-funss 38585  df-funsALTV 38586
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator