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Theorem fnom 8459
Description: Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fnom ยทo Fn (On ร— On)

Proof of Theorem fnom
Dummy variables ๐‘ฅ ๐‘ฆ ๐‘ง are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-omul 8421 . 2 ยทo = (๐‘ฅ โˆˆ On, ๐‘ฆ โˆˆ On โ†ฆ (rec((๐‘ง โˆˆ V โ†ฆ (๐‘ง +o ๐‘ฅ)), โˆ…)โ€˜๐‘ฆ))
2 fvex 6859 . 2 (rec((๐‘ง โˆˆ V โ†ฆ (๐‘ง +o ๐‘ฅ)), โˆ…)โ€˜๐‘ฆ) โˆˆ V
31, 2fnmpoi 8006 1 ยทo Fn (On ร— On)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3447  โˆ…c0 4286   โ†ฆ cmpt 5192   ร— cxp 5635  Oncon0 6321   Fn wfn 6495  โ€˜cfv 6500  (class class class)co 7361  reccrdg 8359   +o coa 8413   ยทo comu 8414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-omul 8421
This theorem is referenced by:  om0x  8469  dmmulpi  10835
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