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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 7612 | . . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V) | |
2 | 1 | rgen 3148 | . . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V |
3 | df-bday 33152 | . . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥) | |
4 | 3 | mptfng 6486 | . . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No ) |
5 | 2, 4 | mpbi 232 | . 2 ⊢ bday Fn No |
6 | 3 | rnmpt 5826 | . . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥} |
7 | noxp1o 33170 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No ) | |
8 | 1oex 8109 | . . . . . . . . 9 ⊢ 1o ∈ V | |
9 | 8 | snnz 4710 | . . . . . . . 8 ⊢ {1o} ≠ ∅ |
10 | dmxp 5798 | . . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦 |
12 | 11 | eqcomi 2830 | . . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o}) |
13 | dmeq 5771 | . . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o})) | |
14 | 13 | rspceeqv 3637 | . . . . . 6 ⊢ (((𝑦 × {1o}) ∈ No ∧ 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
15 | 7, 12, 14 | sylancl 588 | . . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
16 | nodmon 33157 | . . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On) | |
17 | eleq1a 2908 | . . . . . . 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 3280 | . . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On) |
20 | 15, 19 | impbii 211 | . . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥) |
21 | 20 | abbi2i 2953 | . . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥} |
22 | 6, 21 | eqtr4i 2847 | . 2 ⊢ ran bday = On |
23 | df-fo 6360 | . 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 1533 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∅c0 4290 {csn 4566 × cxp 5552 dom cdm 5554 ran crn 5555 Oncon0 6190 Fn wfn 6349 –onto→wfo 6352 1oc1o 8094 No csur 33147 bday cbday 33149 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-1o 8101 df-no 33150 df-bday 33152 |
This theorem is referenced by: nodense 33196 bdayimaon 33197 nosupno 33203 nosupbday 33205 noetalem3 33219 noetalem4 33220 bdayfun 33242 bdayfn 33243 bdaydm 33244 bdayrn 33245 bdayelon 33246 noprc 33249 |
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