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Theorem naddf 8662
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 8656 . 2 +no Fn (On × On)
2 naddcl 8658 . . . 4 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On)
32rgen2 3196 . . 3 𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On
4 fveq2 6877 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = ( +no ‘⟨𝑦, 𝑧⟩))
5 df-ov 7395 . . . . . 6 (𝑦 +no 𝑧) = ( +no ‘⟨𝑦, 𝑧⟩)
64, 5eqtr4di 2789 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = (𝑦 +no 𝑧))
76eleq1d 2817 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On))
87ralxp 5832 . . 3 (∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On)
93, 8mpbir 230 . 2 𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On
10 ffnfv 7101 . 2 ( +no :(On × On)⟶On ↔ ( +no Fn (On × On) ∧ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On))
111, 9, 10mpbir2an 709 1 +no :(On × On)⟶On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wral 3060  cop 4627   × cxp 5666  Oncon0 6352   Fn wfn 6526  wf 6527  cfv 6531  (class class class)co 7392   +no cnadd 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5277  ax-sep 5291  ax-nul 5298  ax-pow 5355  ax-pr 5419  ax-un 7707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3474  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4943  df-iun 4991  df-br 5141  df-opab 5203  df-mpt 5224  df-tr 5258  df-id 5566  df-eprel 5572  df-po 5580  df-so 5581  df-fr 5623  df-se 5624  df-we 5625  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-res 5680  df-ima 5681  df-pred 6288  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6483  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7956  df-2nd 7957  df-frecs 8247  df-nadd 8647
This theorem is referenced by:  naddunif  8674  naddasslem1  8675  naddasslem2  8676
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