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Mirrors > Home > MPE Home > Th. List > Mathboxes > bdayfo | Structured version Visualization version GIF version |
Description: The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.) |
Ref | Expression |
---|---|
bdayfo | ⊢ bday : No –onto→On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexg 7428 | . . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V) | |
2 | 1 | rgen 3098 | . . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V |
3 | df-bday 32679 | . . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥) | |
4 | 3 | mptfng 6317 | . . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No ) |
5 | 2, 4 | mpbi 222 | . 2 ⊢ bday Fn No |
6 | 3 | rnmpt 5670 | . . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥} |
7 | noxp1o 32697 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No ) | |
8 | 1oex 7913 | . . . . . . . . 9 ⊢ 1o ∈ V | |
9 | 8 | snnz 4585 | . . . . . . . 8 ⊢ {1o} ≠ ∅ |
10 | dmxp 5642 | . . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦 |
12 | 11 | eqcomi 2787 | . . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o}) |
13 | dmeq 5622 | . . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o})) | |
14 | 13 | rspceeqv 3553 | . . . . . 6 ⊢ (((𝑦 × {1o}) ∈ No ∧ 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
15 | 7, 12, 14 | sylancl 577 | . . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
16 | nodmon 32684 | . . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On) | |
17 | eleq1a 2861 | . . . . . . 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 3225 | . . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On) |
20 | 15, 19 | impbii 201 | . . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
21 | 20 | abbi2i 2905 | . . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥} |
22 | 6, 21 | eqtr4i 2805 | . 2 ⊢ ran bday = On |
23 | df-fo 6194 | . 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On)) | |
24 | 5, 22, 23 | mpbir2an 698 | 1 ⊢ bday : No –onto→On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 {cab 2758 ≠ wne 2967 ∀wral 3088 ∃wrex 3089 Vcvv 3415 ∅c0 4178 {csn 4441 × cxp 5405 dom cdm 5407 ran crn 5408 Oncon0 6029 Fn wfn 6183 –onto→wfo 6186 1oc1o 7898 No csur 32674 bday cbday 32676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-ord 6032 df-on 6033 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-1o 7905 df-no 32677 df-bday 32679 |
This theorem is referenced by: nodense 32723 bdayimaon 32724 nosupno 32730 nosupbday 32732 noetalem3 32746 noetalem4 32747 bdayfun 32769 bdayfn 32770 bdaydm 32771 bdayrn 32772 bdayelon 32773 noprc 32776 |
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