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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 7607 | . . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V) | |
2 | 1 | rgen 3148 | . . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V |
3 | df-bday 33147 | . . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥) | |
4 | 3 | mptfng 6482 | . . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No ) |
5 | 2, 4 | mpbi 232 | . 2 ⊢ bday Fn No |
6 | 3 | rnmpt 5822 | . . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥} |
7 | noxp1o 33165 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No ) | |
8 | 1oex 8104 | . . . . . . . . 9 ⊢ 1o ∈ V | |
9 | 8 | snnz 4705 | . . . . . . . 8 ⊢ {1o} ≠ ∅ |
10 | dmxp 5794 | . . . . . . . 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 5767 | . . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o})) | |
14 | 13 | rspceeqv 3638 | . . . . . 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 33152 | . . . . . . 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 6356 | . 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 3495 ∅c0 4291 {csn 4561 × cxp 5548 dom cdm 5550 ran crn 5551 Oncon0 6186 Fn wfn 6345 –onto→wfo 6348 1oc1o 8089 No csur 33142 bday cbday 33144 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-1o 8096 df-no 33145 df-bday 33147 |
This theorem is referenced by: nodense 33191 bdayimaon 33192 nosupno 33198 nosupbday 33200 noetalem3 33214 noetalem4 33215 bdayfun 33237 bdayfn 33238 bdaydm 33239 bdayrn 33240 bdayelon 33241 noprc 33244 |
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