ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnoa GIF version

Theorem fnoa 6058
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 6036 . 2 +𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
2 vex 2577 . . 3 𝑦 ∈ V
3 vex 2577 . . . 4 𝑥 ∈ V
4 oafnex 6055 . . . 4 (𝑧 ∈ V ↦ suc 𝑧) Fn V
53, 4rdgexg 6007 . . 3 (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V)
62, 5ax-mp 7 . 2 (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V
71, 6fnmpt2i 5858 1 +𝑜 Fn (On × On)
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574  cmpt 3846  Oncon0 4128  suc csuc 4130   × cxp 4371   Fn wfn 4925  cfv 4930  reccrdg 5987   +𝑜 coa 6029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036
This theorem is referenced by:  dmaddpi  6481
  Copyright terms: Public domain W3C validator