![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > bdayfo | Structured version Visualization version GIF version |
Description: The birthday function maps the surreals onto the ordinals. Axiom B of [Alling] p. 184. (Proof shortened on 14-Apr-2012 by SF). (Contributed by Scott Fenton, 11-Jun-2011.) |
Ref | Expression |
---|---|
bdayfo | ⊢ bday : No –onto→On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexg 7914 | . . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V) | |
2 | 1 | rgen 3053 | . . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V |
3 | df-bday 27674 | . . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥) | |
4 | 3 | mptfng 6700 | . . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No ) |
5 | 2, 4 | mpbi 229 | . 2 ⊢ bday Fn No |
6 | 3 | rnmpt 5961 | . . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥} |
7 | noxp1o 27693 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No ) | |
8 | 1oex 8506 | . . . . . . . . 9 ⊢ 1o ∈ V | |
9 | 8 | snnz 4785 | . . . . . . . 8 ⊢ {1o} ≠ ∅ |
10 | dmxp 5935 | . . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦 |
12 | 11 | eqcomi 2735 | . . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o}) |
13 | dmeq 5910 | . . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o})) | |
14 | 13 | rspceeqv 3630 | . . . . . 6 ⊢ (((𝑦 × {1o}) ∈ No ∧ 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
15 | 7, 12, 14 | sylancl 584 | . . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
16 | nodmon 27680 | . . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On) | |
17 | eleq1a 2821 | . . . . . . 7 ⊢ (dom 𝑥 ∈ On → (𝑦 = dom 𝑥 → 𝑦 ∈ On)) | |
18 | 16, 17 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ No → (𝑦 = dom 𝑥 → 𝑦 ∈ On)) |
19 | 18 | rexlimiv 3138 | . . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On) |
20 | 15, 19 | impbii 208 | . . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
21 | 20 | eqabi 2862 | . . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥} |
22 | 6, 21 | eqtr4i 2757 | . 2 ⊢ ran bday = On |
23 | df-fo 6560 | . 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On)) | |
24 | 5, 22, 23 | mpbir2an 709 | 1 ⊢ bday : No –onto→On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cab 2703 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 Vcvv 3462 ∅c0 4325 {csn 4633 × cxp 5680 dom cdm 5682 ran crn 5683 Oncon0 6376 Fn wfn 6549 –onto→wfo 6552 1oc1o 8489 No csur 27669 bday cbday 27671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-suc 6382 df-fun 6556 df-fn 6557 df-f 6558 df-fo 6560 df-1o 8496 df-no 27672 df-bday 27674 |
This theorem is referenced by: nodense 27722 bdayimaon 27723 nosupno 27733 nosupbday 27735 noinfno 27748 noinfbday 27750 noetasuplem4 27766 noetainflem4 27770 bdayfun 27802 bdayfn 27803 bdaydm 27804 bdayrn 27805 bdayelon 27806 noprc 27809 noeta2 27814 |
Copyright terms: Public domain | W3C validator |