Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elfunsALTV4 Structured version   Visualization version   GIF version

Theorem elfunsALTV4 38676
Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
Assertion
Ref Expression
elfunsALTV4 (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥𝐹 ∈ Rels ))
Distinct variable group:   𝑢,𝐹,𝑥

Proof of Theorem elfunsALTV4
StepHypRef Expression
1 elfunsALTV 38673 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
2 cosselcnvrefrels4 38521 . . . 4 ( ≀ 𝐹 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ ≀ 𝐹 ∈ Rels ))
3 cosselrels 38477 . . . . 5 (𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels )
43biantrud 531 . . . 4 (𝐹 ∈ Rels → (∀𝑢∃*𝑥 𝑢𝐹𝑥 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ ≀ 𝐹 ∈ Rels )))
52, 4bitr4id 290 . . 3 (𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥))
65pm5.32ri 575 . 2 (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥𝐹 ∈ Rels ))
71, 6bitri 275 1 (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥𝐹 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wal 1534  wcel 2105  ∃*wmo 2535   class class class wbr 5147  ccoss 38161   Rels crels 38163   CnvRefRels ccnvrefrels 38169   FunsALTV cfunsALTV 38191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-coss 38392  df-rels 38466  df-ssr 38479  df-cnvrefs 38506  df-cnvrefrels 38507  df-funss 38661  df-funsALTV 38662
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator