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Theorem dffunsALTV5 38688
Description: Alternate definition of the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
Assertion
Ref Expression
dffunsALTV5 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)}
Distinct variable group:   𝑥,𝑓,𝑦

Proof of Theorem dffunsALTV5
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 dffunsALTV4 38687 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2 ineccnvmo2 38361 . . 3 (∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
32rabbii 3442 . 2 {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
41, 3eqtr4i 2768 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)}
Colors of variables: wff setvar class
Syntax hints:  wo 848  wal 1538   = wceq 1540  ∃*wmo 2538  wral 3061  {crab 3436  cin 3950  c0 4333   class class class wbr 5143  ccnv 5684  ran crn 5686  [cec 8743   Rels crels 38184   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-rmo 3380  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-ima 5698  df-ec 8747  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: (None)
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