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Theorem dffunsALTV5 38214
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 38213 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2 ineccnvmo2 37887 . . 3 (∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
32rabbii 3425 . 2 {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
41, 3eqtr4i 2756 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)}
Colors of variables: wff setvar class
Syntax hints:  wo 845  wal 1531   = wceq 1533  ∃*wmo 2526  wral 3051  {crab 3419  cin 3939  c0 4318   class class class wbr 5143  ccnv 5671  ran crn 5673  [cec 8719   Rels crels 37706   FunsALTV cfunsALTV 37734
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8723  df-coss 37938  df-rels 38012  df-ssr 38025  df-cnvrefs 38052  df-cnvrefrels 38053  df-funss 38207  df-funsALTV 38208
This theorem is referenced by: (None)
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