Theorem fnom 8448

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 8414	. 2 ⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
2		fvex 6857	. 2 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦) ∈ V
3	1, 2	fnmpoi 8026	1 ⊢ ·o Fn (On × On)

Syntax hints:  Vcvv 3442  ∅c0 4287   ↦ cmpt 5181   × cxp 5632  Oncon0 6327   Fn wfn 6497  ‘cfv 6502  (class class class)co 7370  reccrdg 8352   +_o coa 8406   ·_o comu 8407

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-omul 8414

This theorem is referenced by:  om0x  8458  dmmulpi  10816