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Theorem dffunsALTV5 38070
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 38069 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2 ineccnvmo2 37742 . . 3 (∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
32rabbii 3432 . 2 {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
41, 3eqtr4i 2757 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)}
Colors of variables: wff setvar class
Syntax hints:  wo 844  wal 1531   = wceq 1533  ∃*wmo 2526  wral 3055  {crab 3426  cin 3942  c0 4317   class class class wbr 5141  ccnv 5668  ran crn 5670  [cec 8703   Rels crels 37558   FunsALTV cfunsALTV 37586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-coss 37794  df-rels 37868  df-ssr 37881  df-cnvrefs 37908  df-cnvrefrels 37909  df-funss 38063  df-funsALTV 38064
This theorem is referenced by: (None)
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