Theorem naddf 8737

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.

Step	Hyp	Ref	Expression
1		naddfn 8731	. 2 ⊢ +no Fn (On × On)
2		naddcl 8733	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On)
3	2	rgen2 3205	. . 3 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On
4		fveq2 6920	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = ( +no ‘⟨𝑦, 𝑧⟩))
5		df-ov 7451	. . . . . 6 ⊢ (𝑦 +no 𝑧) = ( +no ‘⟨𝑦, 𝑧⟩)
6	4, 5	eqtr4di 2798	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = (𝑦 +no 𝑧))
7	6	eleq1d 2829	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On))
8	7	ralxp 5866	. . 3 ⊢ (∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On)
9	3, 8	mpbir 231	. 2 ⊢ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On
10		ffnfv 7153	. 2 ⊢ ( +no :(On × On)⟶On ↔ ( +no Fn (On × On) ∧ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On))
11	1, 9, 10	mpbir2an 710	1 ⊢ +no :(On × On)⟶On

Syntax hints:   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ⟨cop 4654   × cxp 5698  Oncon0 6395   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   +no cnadd 8721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-nadd 8722

This theorem is referenced by:  naddunif  8749  naddasslem1  8750  naddasslem2  8751