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Theorem naddfn 8696
Description: Natural addition is a function over pairs of ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
Assertion
Ref Expression
naddfn +no Fn (On × On)

Proof of Theorem naddfn
Dummy variables 𝑤 𝑎 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nadd 8687 . 2 +no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ {𝑤 ∈ On ∣ ((𝑎 “ ({(1st𝑧)} × (2nd𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1st𝑧) × {(2nd𝑧)})) ⊆ 𝑤)}))
21on2recsfn 8688 1 +no Fn (On × On)
Colors of variables: wff setvar class
Syntax hints:  wa 395  {crab 3429  Vcvv 3471  wss 3947  {csn 4629   cint 4949   × cxp 5676  cima 5681  Oncon0 6369   Fn wfn 6543  cfv 6548  cmpo 7422  1st c1st 7991  2nd c2nd 7992   +no cnadd 8686
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 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-1st 7993  df-2nd 7994  df-frecs 8287  df-nadd 8687
This theorem is referenced by:  naddcllem  8697  naddov2  8700  naddf  8702  naddunif  8714  naddasslem1  8715  naddasslem2  8716
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