MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noetainflem2 Structured version   Visualization version   GIF version

Theorem noetainflem2 27102
Description: Lemma for noeta 27107. The restriction of 𝑊 to the domain of 𝑇 is 𝑇. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypotheses
Ref Expression
noetainflem.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
noetainflem.2 𝑊 = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
Assertion
Ref Expression
noetainflem2 ((𝐵 No 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)
Distinct variable group:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetainflem2
StepHypRef Expression
1 noetainflem.2 . . . 4 𝑊 = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
21reseq1i 5934 . . 3 (𝑊 ↾ dom 𝑇) = ((𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) ↾ dom 𝑇)
3 resundir 5953 . . 3 ((𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) ↾ dom 𝑇) = ((𝑇 ↾ dom 𝑇) ∪ (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇))
42, 3eqtri 2761 . 2 (𝑊 ↾ dom 𝑇) = ((𝑇 ↾ dom 𝑇) ∪ (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇))
5 noetainflem.1 . . . . . . 7 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
65noinfno 27082 . . . . . 6 ((𝐵 No 𝐵 ∈ V) → 𝑇 No )
7 nofun 27013 . . . . . 6 (𝑇 No → Fun 𝑇)
86, 7syl 17 . . . . 5 ((𝐵 No 𝐵 ∈ V) → Fun 𝑇)
9 funrel 6519 . . . . 5 (Fun 𝑇 → Rel 𝑇)
10 resdm 5983 . . . . 5 (Rel 𝑇 → (𝑇 ↾ dom 𝑇) = 𝑇)
118, 9, 103syl 18 . . . 4 ((𝐵 No 𝐵 ∈ V) → (𝑇 ↾ dom 𝑇) = 𝑇)
12 dmres 5960 . . . . . . 7 dom (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = (dom 𝑇 ∩ dom ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
13 2oex 8424 . . . . . . . . . . 11 2o ∈ V
1413snnz 4738 . . . . . . . . . 10 {2o} ≠ ∅
15 dmxp 5885 . . . . . . . . . 10 ({2o} ≠ ∅ → dom ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) = (suc ( bday 𝐴) ∖ dom 𝑇))
1614, 15ax-mp 5 . . . . . . . . 9 dom ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) = (suc ( bday 𝐴) ∖ dom 𝑇)
1716ineq2i 4170 . . . . . . . 8 (dom 𝑇 ∩ dom ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) = (dom 𝑇 ∩ (suc ( bday 𝐴) ∖ dom 𝑇))
18 disjdif 4432 . . . . . . . 8 (dom 𝑇 ∩ (suc ( bday 𝐴) ∖ dom 𝑇)) = ∅
1917, 18eqtri 2761 . . . . . . 7 (dom 𝑇 ∩ dom ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) = ∅
2012, 19eqtri 2761 . . . . . 6 dom (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
21 relres 5967 . . . . . . 7 Rel (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)
22 reldm0 5884 . . . . . . 7 (Rel (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) → ((((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅ ↔ dom (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅))
2321, 22ax-mp 5 . . . . . 6 ((((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅ ↔ dom (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅)
2420, 23mpbir 230 . . . . 5 (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
2524a1i 11 . . . 4 ((𝐵 No 𝐵 ∈ V) → (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅)
2611, 25uneq12d 4125 . . 3 ((𝐵 No 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = (𝑇 ∪ ∅))
27 un0 4351 . . 3 (𝑇 ∪ ∅) = 𝑇
2826, 27eqtrdi 2789 . 2 ((𝐵 No 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = 𝑇)
294, 28eqtrid 2785 1 ((𝐵 No 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wne 2940  wral 3061  wrex 3070  Vcvv 3444  cdif 3908  cun 3909  cin 3910  wss 3911  c0 4283  ifcif 4487  {csn 4587  cop 4593   cuni 4866   class class class wbr 5106  cmpt 5189   × cxp 5632  dom cdm 5634  cres 5636  cima 5637  Rel wrel 5639  suc csuc 6320  cio 6447  Fun wfun 6491  cfv 6497  crio 7313  1oc1o 8406  2oc2o 8407   No csur 27004   <s cslt 27005   bday cbday 27006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009
This theorem is referenced by:  noetainflem3  27103  noetainflem4  27104  noetalem1  27105
  Copyright terms: Public domain W3C validator