Theorem fnom 6713

Description: Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)

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 6682	. 2 ⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
2		vex 2824	. . 3 ⊢ 𝑦 ∈ V
3		0ex 4255	. . . 4 ⊢ ∅ ∈ V
4		vex 2824	. . . . 5 ⊢ 𝑥 ∈ V
5		omfnex 6712	. . . . 5 ⊢ (𝑥 ∈ V → (𝑧 ∈ V ↦ (𝑧 +o 𝑥)) Fn V)
6	4, 5	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ V ↦ (𝑧 +o 𝑥)) Fn V
7	3, 6	rdgexg 6650	. . 3 ⊢ (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦) ∈ V)
8	2, 7	ax-mp 5	. 2 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦) ∈ V
9	1, 8	fnmpoi 6429	1 ⊢ ·o Fn (On × On)

Syntax hints:   ∈ wcel 2209  Vcvv 2821  ∅c0 3520   ↦ cmpt 4187  Oncon0 4503   × cxp 4767   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075  reccrdg 6630   +_o coa 6674   ·_o comu 6675

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682

This theorem is referenced by:  dmmulpi  7683