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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 8093 | . 2 ⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦)) | |
2 | fvex 6669 | . 2 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦) ∈ V | |
3 | 1, 2 | fnmpoi 7754 | 1 ⊢ ·o Fn (On × On) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3486 ∅c0 4279 ↦ cmpt 5132 × cxp 5539 Oncon0 6177 Fn wfn 6336 ‘cfv 6341 (class class class)co 7142 reccrdg 8031 +o coa 8085 ·o comu 8086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-omul 8093 |
This theorem is referenced by: om0x 8130 dmmulpi 10299 |
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