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Theorem fnoa 6462
Description: Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.)
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 6435 . 2 +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
2 vex 2752 . . 3 𝑦 ∈ V
3 vex 2752 . . . 4 𝑥 ∈ V
4 oafnex 6459 . . . 4 (𝑧 ∈ V ↦ suc 𝑧) Fn V
53, 4rdgexg 6404 . . 3 (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V)
62, 5ax-mp 5 . 2 (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V
71, 6fnmpoi 6219 1 +o Fn (On × On)
Colors of variables: wff set class
Syntax hints:  wcel 2158  Vcvv 2749  cmpt 4076  Oncon0 4375  suc csuc 4377   × cxp 4636   Fn wfn 5223  cfv 5228  reccrdg 6384   +o coa 6428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435
This theorem is referenced by:  dmaddpi  7338
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