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Theorem erdszelem5 34714
Description: Lemma for erdsze 34721. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n (šœ‘ → š‘ ∈ ā„•)
erdsze.f (šœ‘ → š¹:(1...š‘)–1-1ā†’ā„)
erdszelem.k š¾ = (š‘„ ∈ (1...š‘) ↦ sup((♯ ā€œ {š‘¦ ∈ š’« (1...š‘„) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š‘„ ∈ š‘¦)}), ā„, < ))
erdszelem.o š‘‚ Or ā„
Assertion
Ref Expression
erdszelem5 ((šœ‘ ∧ š“ ∈ (1...š‘)) → (š¾ā€˜š“) ∈ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}))
Distinct variable groups:   š‘„,š‘¦,š¹   š‘„,š“,š‘¦   š‘„,š‘‚,š‘¦   š‘„,š‘,š‘¦   šœ‘,š‘„,š‘¦
Allowed substitution hints:   š¾(š‘„,š‘¦)

Proof of Theorem erdszelem5
StepHypRef Expression
1 erdsze.n . . . 4 (šœ‘ → š‘ ∈ ā„•)
2 erdsze.f . . . 4 (šœ‘ → š¹:(1...š‘)–1-1ā†’ā„)
3 erdszelem.k . . . 4 š¾ = (š‘„ ∈ (1...š‘) ↦ sup((♯ ā€œ {š‘¦ ∈ š’« (1...š‘„) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š‘„ ∈ š‘¦)}), ā„, < ))
41, 2, 3erdszelem3 34712 . . 3 (š“ ∈ (1...š‘) → (š¾ā€˜š“) = sup((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}), ā„, < ))
54adantl 481 . 2 ((šœ‘ ∧ š“ ∈ (1...š‘)) → (š¾ā€˜š“) = sup((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}), ā„, < ))
6 snex 5424 . . . . . 6 {š“} ∈ V
7 hashf 14303 . . . . . . 7 ♯:V⟶(ā„•0 ∪ {+āˆž})
87fdmi 6723 . . . . . 6 dom ♯ = V
96, 8eleqtrri 2826 . . . . 5 {š“} ∈ dom ♯
10 erdszelem.o . . . . . 6 š‘‚ Or ā„
111, 2, 3, 10erdszelem4 34713 . . . . 5 ((šœ‘ ∧ š“ ∈ (1...š‘)) → {š“} ∈ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)})
12 inelcm 4459 . . . . 5 (({š“} ∈ dom ♯ ∧ {š“} ∈ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) → (dom ♯ ∩ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ…)
139, 11, 12sylancr 586 . . . 4 ((šœ‘ ∧ š“ ∈ (1...š‘)) → (dom ♯ ∩ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ…)
14 imadisj 6073 . . . . 5 ((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) = āˆ… ↔ (dom ♯ ∩ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) = āˆ…)
1514necon3bii 2987 . . . 4 ((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ… ↔ (dom ♯ ∩ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ…)
1613, 15sylibr 233 . . 3 ((šœ‘ ∧ š“ ∈ (1...š‘)) → (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ…)
17 eqid 2726 . . . . . 6 {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)} = {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}
1817erdszelem2 34711 . . . . 5 ((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ∈ Fin ∧ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) āŠ† ā„•)
1918simpli 483 . . . 4 (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ∈ Fin
2018simpri 485 . . . . 5 (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) āŠ† ā„•
21 nnssre 12220 . . . . 5 ā„• āŠ† ā„
2220, 21sstri 3986 . . . 4 (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) āŠ† ā„
23 ltso 11298 . . . . 5 < Or ā„
24 fisupcl 9466 . . . . 5 (( < Or ā„ ∧ ((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ∈ Fin ∧ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ… ∧ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) āŠ† ā„)) → sup((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}), ā„, < ) ∈ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}))
2523, 24mpan 687 . . . 4 (((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ∈ Fin ∧ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ… ∧ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) āŠ† ā„) → sup((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}), ā„, < ) ∈ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}))
2619, 22, 25mp3an13 1448 . . 3 ((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}) ≠ āˆ… → sup((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}), ā„, < ) ∈ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}))
2716, 26syl 17 . 2 ((šœ‘ ∧ š“ ∈ (1...š‘)) → sup((♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}), ā„, < ) ∈ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}))
285, 27eqeltrd 2827 1 ((šœ‘ ∧ š“ ∈ (1...š‘)) → (š¾ā€˜š“) ∈ (♯ ā€œ {š‘¦ ∈ š’« (1...š“) ∣ ((š¹ ↾ š‘¦) Isom < , š‘‚ (š‘¦, (š¹ ā€œ š‘¦)) ∧ š“ ∈ š‘¦)}))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  {crab 3426  Vcvv 3468   ∪ cun 3941   ∩ cin 3942   āŠ† wss 3943  āˆ…c0 4317  š’« cpw 4597  {csn 4623   ↦ cmpt 5224   Or wor 5580  dom cdm 5669   ↾ cres 5671   ā€œ cima 5672  ā€“1-1→wf1 6534  ā€˜cfv 6537   Isom wiso 6538  (class class class)co 7405  Fincfn 8941  supcsup 9437  ā„cr 11111  1c1 11113  +āˆžcpnf 11249   < clt 11252  ā„•cn 12216  ā„•0cn0 12476  ...cfz 13490  ā™Æchash 14295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13491  df-hash 14296
This theorem is referenced by:  erdszelem6  34715  erdszelem7  34716  erdszelem8  34717
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