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Theorem fnoa 7540
 Description: Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fnoa +𝑜 Fn (On × On)

Proof of Theorem fnoa
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-oadd 7516 . 2 +𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
2 fvex 6163 . 2 (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V
31, 2fnmpt2i 7191 1 +𝑜 Fn (On × On)
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3189   ↦ cmpt 4678   × cxp 5077  Oncon0 5687  suc csuc 5689   Fn wfn 5847  ‘cfv 5852  reccrdg 7457   +𝑜 coa 7509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-oadd 7516 This theorem is referenced by:  cantnfvalf  8514  dmaddpi  9664
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