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Theorem fnoa 8124
 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 8097 . 2 +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
2 fvex 6680 . 2 (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V
31, 2fnmpoi 7759 1 +o Fn (On × On)
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3500   ↦ cmpt 5143   × cxp 5552  Oncon0 6189  suc csuc 6191   Fn wfn 6347  ‘cfv 6352  reccrdg 8036   +o coa 8090 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-oadd 8097 This theorem is referenced by:  cantnfvalf  9117  dmaddpi  10301
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