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Theorem naddf 8718
Description: Function statement for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
Assertion
Ref Expression
naddf +no :(On × On)⟶On

Proof of Theorem naddf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 naddfn 8712 . 2 +no Fn (On × On)
2 naddcl 8714 . . . 4 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On)
32rgen2 3197 . . 3 𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On
4 fveq2 6907 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = ( +no ‘⟨𝑦, 𝑧⟩))
5 df-ov 7434 . . . . . 6 (𝑦 +no 𝑧) = ( +no ‘⟨𝑦, 𝑧⟩)
64, 5eqtr4di 2793 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = (𝑦 +no 𝑧))
76eleq1d 2824 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On))
87ralxp 5855 . . 3 (∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On)
93, 8mpbir 231 . 2 𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On
10 ffnfv 7139 . 2 ( +no :(On × On)⟶On ↔ ( +no Fn (On × On) ∧ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On))
111, 9, 10mpbir2an 711 1 +no :(On × On)⟶On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  cop 4637   × cxp 5687  Oncon0 6386   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431   +no cnadd 8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-nadd 8703
This theorem is referenced by:  naddunif  8730  naddasslem1  8731  naddasslem2  8732
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