Theorem fnom 8508

Description: Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
fnom	⊢ ·o Fn (On × On)

Proof of Theorem fnom

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-omul 8470	. 2 ⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
2		fvex 6904	. 2 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦) ∈ V
3	1, 2	fnmpoi 8055	1 ⊢ ·o Fn (On × On)

Syntax hints:  Vcvv 3474  ∅c0 4322   ↦ cmpt 5231   × cxp 5674  Oncon0 6364   Fn wfn 6538  ‘cfv 6543  (class class class)co 7408  reccrdg 8408   +_o coa 8462   ·_o comu 8463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-omul 8470

This theorem is referenced by:  om0x  8518  dmmulpi  10885