![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fnoa | Structured version Visualization version GIF version |
Description: Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
fnoa | ⊢ +o Fn (On × On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oadd 8420 | . 2 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)) | |
2 | fvex 6859 | . 2 ⊢ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V | |
3 | 1, 2 | fnmpoi 8006 | 1 ⊢ +o Fn (On × On) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3447 ↦ cmpt 5192 × cxp 5635 Oncon0 6321 suc csuc 6323 Fn wfn 6495 ‘cfv 6500 reccrdg 8359 +o coa 8413 |
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-oadd 8420 |
This theorem is referenced by: cantnfvalf 9609 dmaddpi 10834 |
Copyright terms: Public domain | W3C validator |