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Theorem nosupdm 32226
Description: The domain of the surreal supremum when there is no maximum. The primary point of this theorem is to change bound variable. (Contributed by Scott Fenton, 6-Dec-2021.)
Hypothesis
Ref Expression
nosupdm.1 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
nosupdm (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
Distinct variable groups:   𝐴,𝑔   𝐴,𝑝,𝑞,𝑢,𝑣,𝑦,𝑧   𝑢,𝑔,𝑣,𝑦   𝑞,𝑝,𝑢,𝑣,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑔,𝑞,𝑝)

Proof of Theorem nosupdm
StepHypRef Expression
1 nosupdm.1 . . . . 5 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
2 iffalse 4252 . . . . 5 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
31, 2syl5eq 2811 . . . 4 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
43dmeqd 5494 . . 3 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
5 iotaex 6048 . . . 4 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
6 eqid 2765 . . . 4 (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
75, 6dmmpti 6201 . . 3 dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
84, 7syl6eq 2815 . 2 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
9 dmeq 5492 . . . . . . 7 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
109eleq2d 2830 . . . . . 6 (𝑢 = 𝑝 → (𝑦 ∈ dom 𝑢𝑦 ∈ dom 𝑝))
11 breq1 4812 . . . . . . . . . 10 (𝑣 = 𝑞 → (𝑣 <s 𝑢𝑞 <s 𝑢))
1211notbid 309 . . . . . . . . 9 (𝑣 = 𝑞 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑞 <s 𝑢))
13 reseq1 5559 . . . . . . . . . 10 (𝑣 = 𝑞 → (𝑣 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))
1413eqeq2d 2775 . . . . . . . . 9 (𝑣 = 𝑞 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)))
1512, 14imbi12d 335 . . . . . . . 8 (𝑣 = 𝑞 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑞 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))))
1615cbvralv 3319 . . . . . . 7 (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑞𝐴𝑞 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)))
17 breq2 4813 . . . . . . . . . 10 (𝑢 = 𝑝 → (𝑞 <s 𝑢𝑞 <s 𝑝))
1817notbid 309 . . . . . . . . 9 (𝑢 = 𝑝 → (¬ 𝑞 <s 𝑢 ↔ ¬ 𝑞 <s 𝑝))
19 reseq1 5559 . . . . . . . . . 10 (𝑢 = 𝑝 → (𝑢 ↾ suc 𝑦) = (𝑝 ↾ suc 𝑦))
2019eqeq1d 2767 . . . . . . . . 9 (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦) ↔ (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)))
2118, 20imbi12d 335 . . . . . . . 8 (𝑢 = 𝑝 → ((¬ 𝑞 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)) ↔ (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))))
2221ralbidv 3133 . . . . . . 7 (𝑢 = 𝑝 → (∀𝑞𝐴𝑞 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)) ↔ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))))
2316, 22syl5bb 274 . . . . . 6 (𝑢 = 𝑝 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))))
2410, 23anbi12d 624 . . . . 5 (𝑢 = 𝑝 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑦 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)))))
2524cbvrexv 3320 . . . 4 (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑝𝐴 (𝑦 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))))
26 eleq1w 2827 . . . . . 6 (𝑦 = 𝑧 → (𝑦 ∈ dom 𝑝𝑧 ∈ dom 𝑝))
27 suceq 5973 . . . . . . . . . 10 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
2827reseq2d 5565 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑝 ↾ suc 𝑦) = (𝑝 ↾ suc 𝑧))
2927reseq2d 5565 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑞 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑧))
3028, 29eqeq12d 2780 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦) ↔ (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))
3130imbi2d 331 . . . . . . 7 (𝑦 = 𝑧 → ((¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)) ↔ (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
3231ralbidv 3133 . . . . . 6 (𝑦 = 𝑧 → (∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦)) ↔ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
3326, 32anbi12d 624 . . . . 5 (𝑦 = 𝑧 → ((𝑦 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))) ↔ (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))))
3433rexbidv 3199 . . . 4 (𝑦 = 𝑧 → (∃𝑝𝐴 (𝑦 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑦) = (𝑞 ↾ suc 𝑦))) ↔ ∃𝑝𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))))
3525, 34syl5bb 274 . . 3 (𝑦 = 𝑧 → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑝𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))))
3635cbvabv 2890 . 2 {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} = {𝑧 ∣ ∃𝑝𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}
378, 36syl6eq 2815 1 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  cun 3730  ifcif 4243  {csn 4334  cop 4340   class class class wbr 4809  cmpt 4888  dom cdm 5277  cres 5279  suc csuc 5910  cio 6029  cfv 6068  crio 6802  2𝑜c2o 7758   <s cslt 32170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071
This theorem is referenced by:  nosupbnd1lem3  32232  nosupbnd1lem5  32234  nosupbnd2  32238
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