Theorem naddf 8671

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 8664	. 2 ⊢ +no Fn (On × On)
2		naddcl 8666	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On)
3	2	rgen2 3202	. . 3 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On
4		fveq2 6879	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = ( +no ‘⟨𝑦, 𝑧⟩))
5		df-ov 7417	. . . . . 6 ⊢ (𝑦 +no 𝑧) = ( +no ‘⟨𝑦, 𝑧⟩)
6	4, 5	eqtr4di 2813	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +no ‘𝑥) = (𝑦 +no 𝑧))
7	6	eleq1d 2845	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On))
8	7	ralxp 5821	. . 3 ⊢ (∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On)
9	3, 8	mpbir 234	. 2 ⊢ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On
10		ffnfv 7113	. 2 ⊢ ( +no :(On × On)⟶On ↔ ( +no Fn (On × On) ∧ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On))
11	1, 9, 10	mpbir2an 724	1 ⊢ +no :(On × On)⟶On

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ⟨cop 4590   × cxp 5653  Oncon0 6357   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   +no cnadd 8654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-frecs 8281  df-nadd 8655

This theorem is used by:  naddunif  8683  naddasslem1  8684  naddasslem2  8685