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Theorem dffunsALTV4 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∃*𝑦1𝑥1𝑓𝑦1}. (Contributed by Peter Mazsa, 31-Aug-2021.)
Assertion
Ref Expression
dffunsALTV4 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
Distinct variable group:   𝑢,𝑓,𝑥

Proof of Theorem dffunsALTV4
StepHypRef Expression
1 dffunsALTV 38684 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2 cosselcnvrefrels4 38541 . . 3 ( ≀ 𝑓 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels ))
3 cosselrels 38497 . . . 4 (𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
43biantrud 531 . . 3 (𝑓 ∈ Rels → (∀𝑢∃*𝑥 𝑢𝑓𝑥 ↔ (∀𝑢∃*𝑥 𝑢𝑓𝑥 ∧ ≀ 𝑓 ∈ Rels )))
52, 4bitr4id 290 . 2 (𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥))
61, 5rabimbieq 38252 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
Colors of variables: wff setvar class
Syntax hints:  wa 395  wal 1538   = wceq 1540  wcel 2108  ∃*wmo 2538  {crab 3436   class class class wbr 5143  ccoss 38182   Rels crels 38184   CnvRefRels ccnvrefrels 38190   FunsALTV cfunsALTV 38212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-coss 38412  df-rels 38486  df-ssr 38499  df-cnvrefs 38526  df-cnvrefrels 38527  df-funss 38681  df-funsALTV 38682
This theorem is referenced by:  dffunsALTV5  38688
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