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Theorem bnj1523 30393
Description: Technical lemma for bnj1522 30394. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1523.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1523.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1523.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1523.4 𝐹 = 𝐶
bnj1523.5 (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
bnj1523.6 (𝜓 ↔ (𝜑𝐹𝐻))
bnj1523.7 (𝜒 ↔ (𝜓𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))
bnj1523.8 𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}
bnj1523.9 (𝜃 ↔ (𝜒𝑦𝐷 ∧ ∀𝑧𝐷 ¬ 𝑧𝑅𝑦))
Assertion
Ref Expression
bnj1523 ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝑦,𝐴,𝑧,𝑥   𝐵,𝑓   𝑦,𝐷,𝑧   𝑦,𝐹,𝑧   𝐺,𝑑,𝑓,𝑥   𝑦,𝐺   𝑥,𝐻,𝑦,𝑧   𝑅,𝑑,𝑓,𝑥   𝑦,𝑅,𝑧   𝑌,𝑑   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑧,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝐺(𝑧)   𝐻(𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓)

Proof of Theorem bnj1523
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1523.5 . 2 (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
2 bnj1523.6 . . 3 (𝜓 ↔ (𝜑𝐹𝐻))
3 bnj1523.9 . . . . . . . . . . . . 13 (𝜃 ↔ (𝜒𝑦𝐷 ∧ ∀𝑧𝐷 ¬ 𝑧𝑅𝑦))
4 bnj1523.7 . . . . . . . . . . . . . 14 (𝜒 ↔ (𝜓𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))
5 bnj1523.1 . . . . . . . . . . . . . . . . 17 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
6 bnj1523.2 . . . . . . . . . . . . . . . . 17 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
7 bnj1523.3 . . . . . . . . . . . . . . . . 17 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
8 bnj1523.4 . . . . . . . . . . . . . . . . 17 𝐹 = 𝐶
95, 6, 7, 8bnj60 30384 . . . . . . . . . . . . . . . 16 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
101, 9bnj835 30083 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝐴)
112, 10bnj832 30082 . . . . . . . . . . . . . 14 (𝜓𝐹 Fn 𝐴)
124, 11bnj835 30083 . . . . . . . . . . . . 13 (𝜒𝐹 Fn 𝐴)
133, 12bnj835 30083 . . . . . . . . . . . 12 (𝜃𝐹 Fn 𝐴)
141simp2bi 1070 . . . . . . . . . . . . . . 15 (𝜑𝐻 Fn 𝐴)
152, 14bnj832 30082 . . . . . . . . . . . . . 14 (𝜓𝐻 Fn 𝐴)
164, 15bnj835 30083 . . . . . . . . . . . . 13 (𝜒𝐻 Fn 𝐴)
173, 16bnj835 30083 . . . . . . . . . . . 12 (𝜃𝐻 Fn 𝐴)
18 bnj213 30206 . . . . . . . . . . . . 13 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1918a1i 11 . . . . . . . . . . . 12 (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
203simp3bi 1071 . . . . . . . . . . . . . . . . 17 (𝜃 → ∀𝑧𝐷 ¬ 𝑧𝑅𝑦)
2120bnj1211 30122 . . . . . . . . . . . . . . . 16 (𝜃 → ∀𝑧(𝑧𝐷 → ¬ 𝑧𝑅𝑦))
22 con2b 348 . . . . . . . . . . . . . . . . 17 ((𝑧𝐷 → ¬ 𝑧𝑅𝑦) ↔ (𝑧𝑅𝑦 → ¬ 𝑧𝐷))
2322albii 1737 . . . . . . . . . . . . . . . 16 (∀𝑧(𝑧𝐷 → ¬ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷))
2421, 23sylib 207 . . . . . . . . . . . . . . 15 (𝜃 → ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷))
25 bnj1418 30362 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧𝑅𝑦)
2625imim1i 61 . . . . . . . . . . . . . . . 16 ((𝑧𝑅𝑦 → ¬ 𝑧𝐷) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2726alimi 1730 . . . . . . . . . . . . . . 15 (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2824, 27syl 17 . . . . . . . . . . . . . 14 (𝜃 → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2928bnj1142 30114 . . . . . . . . . . . . 13 (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅) ¬ 𝑧𝐷)
30 bnj1523.8 . . . . . . . . . . . . . 14 𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}
315bnj1309 30344 . . . . . . . . . . . . . . . . . . 19 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
327, 31bnj1307 30345 . . . . . . . . . . . . . . . . . 18 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3332nfcii 2742 . . . . . . . . . . . . . . . . 17 𝑥𝐶
3433nfuni 4378 . . . . . . . . . . . . . . . 16 𝑥 𝐶
358, 34nfcxfr 2749 . . . . . . . . . . . . . . 15 𝑥𝐹
3635nfcrii 2744 . . . . . . . . . . . . . 14 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
3730, 36bnj1534 30177 . . . . . . . . . . . . 13 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
3829, 18, 37bnj1533 30176 . . . . . . . . . . . 12 (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅)(𝐹𝑧) = (𝐻𝑧))
3913, 17, 19, 38bnj1536 30178 . . . . . . . . . . 11 (𝜃 → (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) = (𝐻 ↾ pred(𝑦, 𝐴, 𝑅)))
4039opeq2d 4347 . . . . . . . . . 10 (𝜃 → ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
4140fveq2d 6107 . . . . . . . . 9 (𝜃 → (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
425, 6, 7, 8bnj1500 30390 . . . . . . . . . . . . . . 15 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
431, 42bnj835 30083 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
442, 43bnj832 30082 . . . . . . . . . . . . 13 (𝜓 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
454, 44bnj835 30083 . . . . . . . . . . . 12 (𝜒 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
4645, 36bnj1529 30392 . . . . . . . . . . 11 (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
473, 46bnj835 30083 . . . . . . . . . 10 (𝜃 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
4830bnj21 30037 . . . . . . . . . . 11 𝐷𝐴
493simp2bi 1070 . . . . . . . . . . 11 (𝜃𝑦𝐷)
5048, 49bnj1213 30123 . . . . . . . . . 10 (𝜃𝑦𝐴)
5147, 50bnj1294 30142 . . . . . . . . 9 (𝜃 → (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
521simp3bi 1071 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
532, 52bnj832 30082 . . . . . . . . . . . . 13 (𝜓 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
544, 53bnj835 30083 . . . . . . . . . . . 12 (𝜒 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55 ax-5 1827 . . . . . . . . . . . 12 (𝑣𝐻 → ∀𝑥 𝑣𝐻)
5654, 55bnj1529 30392 . . . . . . . . . . 11 (𝜒 → ∀𝑦𝐴 (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
573, 56bnj835 30083 . . . . . . . . . 10 (𝜃 → ∀𝑦𝐴 (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
5857, 50bnj1294 30142 . . . . . . . . 9 (𝜃 → (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
5941, 51, 583eqtr4d 2654 . . . . . . . 8 (𝜃 → (𝐹𝑦) = (𝐻𝑦))
6030, 36bnj1534 30177 . . . . . . . . . . 11 𝐷 = {𝑦𝐴 ∣ (𝐹𝑦) ≠ (𝐻𝑦)}
6160bnj1538 30179 . . . . . . . . . 10 (𝑦𝐷 → (𝐹𝑦) ≠ (𝐻𝑦))
623, 61bnj836 30084 . . . . . . . . 9 (𝜃 → (𝐹𝑦) ≠ (𝐻𝑦))
6362neneqd 2787 . . . . . . . 8 (𝜃 → ¬ (𝐹𝑦) = (𝐻𝑦))
6459, 63pm2.65i 184 . . . . . . 7 ¬ 𝜃
6564nex 1722 . . . . . 6 ¬ ∃𝑦𝜃
661simp1bi 1069 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
672, 66bnj832 30082 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
684, 67bnj835 30083 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
6948a1i 11 . . . . . . . 8 (𝜒𝐷𝐴)
704simp2bi 1070 . . . . . . . . . 10 (𝜒𝑥𝐴)
714simp3bi 1071 . . . . . . . . . 10 (𝜒 → (𝐹𝑥) ≠ (𝐻𝑥))
7230rabeq2i 3170 . . . . . . . . . 10 (𝑥𝐷 ↔ (𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))
7370, 71, 72sylanbrc 695 . . . . . . . . 9 (𝜒𝑥𝐷)
74 ne0i 3880 . . . . . . . . 9 (𝑥𝐷𝐷 ≠ ∅)
7573, 74syl 17 . . . . . . . 8 (𝜒𝐷 ≠ ∅)
76 bnj69 30332 . . . . . . . 8 ((𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅) → ∃𝑦𝐷𝑧𝐷 ¬ 𝑧𝑅𝑦)
7768, 69, 75, 76syl3anc 1318 . . . . . . 7 (𝜒 → ∃𝑦𝐷𝑧𝐷 ¬ 𝑧𝑅𝑦)
7877, 3bnj1209 30121 . . . . . 6 (𝜒 → ∃𝑦𝜃)
7965, 78mto 187 . . . . 5 ¬ 𝜒
8079nex 1722 . . . 4 ¬ ∃𝑥𝜒
812simprbi 479 . . . . . 6 (𝜓𝐹𝐻)
8211, 15, 81, 36bnj1542 30181 . . . . 5 (𝜓 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐻𝑥))
835, 6, 7, 8, 1, 2bnj1525 30391 . . . . 5 (𝜓 → ∀𝑥𝜓)
8482, 4, 83bnj1521 30175 . . . 4 (𝜓 → ∃𝑥𝜒)
8580, 84mto 187 . . 3 ¬ 𝜓
862, 85bnj1541 30180 . 2 (𝜑𝐹 = 𝐻)
871, 86sylbir 224 1 ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wne 2780  wral 2896  wrex 2897  {crab 2900  wss 3540  c0 3874  cop 4131   cuni 4372   class class class wbr 4583  cres 5040   Fn wfn 5799  cfv 5804   predc-bnj14 30007   FrSe w-bnj15 30011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016
This theorem is referenced by:  bnj1522  30394
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