Theorem naddf 8645

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 8639	. 2 ⊢ +no Fn (On × On)
2		naddcl 8641	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On)
3	2	rgen2 3177	. . 3 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On
4		fveq2 6858	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = ( +no ‘⟨𝑦, 𝑧⟩))
5		df-ov 7390	. . . . . 6 ⊢ (𝑦 +no 𝑧) = ( +no ‘⟨𝑦, 𝑧⟩)
6	4, 5	eqtr4di 2782	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = (𝑦 +no 𝑧))
7	6	eleq1d 2813	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On))
8	7	ralxp 5805	. . 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 7091	. 2 ⊢ ( +no :(On × On)⟶On ↔ ( +no Fn (On × On) ∧ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On))
11	1, 9, 10	mpbir2an 711	1 ⊢ +no :(On × On)⟶On

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4595   × cxp 5636  Oncon0 6332   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   +no cnadd 8629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-frecs 8260  df-nadd 8630

This theorem is referenced by:  naddunif  8657  naddasslem1  8658  naddasslem2  8659