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Mirrors > Home > MPE Home > Th. List > fnom | Structured version Visualization version GIF version |
Description: Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
fnom | ⊢ ·o Fn (On × On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-omul 8101 | . 2 ⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦)) | |
2 | fvex 6678 | . 2 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦) ∈ V | |
3 | 1, 2 | fnmpoi 7762 | 1 ⊢ ·o Fn (On × On) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3495 ∅c0 4291 ↦ cmpt 5139 × cxp 5548 Oncon0 6186 Fn wfn 6345 ‘cfv 6350 (class class class)co 7150 reccrdg 8039 +o coa 8093 ·o comu 8094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-omul 8101 |
This theorem is referenced by: om0x 8138 dmmulpi 10307 |
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