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Theorem dffunsALTV5 38096
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 38095 . 2 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
2 ineccnvmo2 37768 . . 3 (∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥)
32rabbii 3433 . 2 {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
41, 3eqtr4i 2758 1 FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)}
Colors of variables: wff setvar class
Syntax hints:  wo 846  wal 1532   = wceq 1534  ∃*wmo 2527  wral 3056  {crab 3427  cin 3943  c0 4318   class class class wbr 5142  ccnv 5671  ran crn 5673  [cec 8716   Rels crels 37585   FunsALTV cfunsALTV 37613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rmo 3371  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  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 8720  df-coss 37820  df-rels 37894  df-ssr 37907  df-cnvrefs 37934  df-cnvrefrels 37935  df-funss 38089  df-funsALTV 38090
This theorem is referenced by: (None)
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