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Theorem bnj1523 31587
Description: Technical lemma for bnj1522 31588. 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 31578 . . . . . . . . . . . . . . . 16 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
101, 9bnj835 31277 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝐴)
112, 10bnj832 31276 . . . . . . . . . . . . . 14 (𝜓𝐹 Fn 𝐴)
124, 11bnj835 31277 . . . . . . . . . . . . 13 (𝜒𝐹 Fn 𝐴)
133, 12bnj835 31277 . . . . . . . . . . . 12 (𝜃𝐹 Fn 𝐴)
141simp2bi 1176 . . . . . . . . . . . . . . 15 (𝜑𝐻 Fn 𝐴)
152, 14bnj832 31276 . . . . . . . . . . . . . 14 (𝜓𝐻 Fn 𝐴)
164, 15bnj835 31277 . . . . . . . . . . . . 13 (𝜒𝐻 Fn 𝐴)
173, 16bnj835 31277 . . . . . . . . . . . 12 (𝜃𝐻 Fn 𝐴)
18 bnj213 31400 . . . . . . . . . . . . 13 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1918a1i 11 . . . . . . . . . . . 12 (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
203simp3bi 1177 . . . . . . . . . . . . . . . . 17 (𝜃 → ∀𝑧𝐷 ¬ 𝑧𝑅𝑦)
2120bnj1211 31316 . . . . . . . . . . . . . . . 16 (𝜃 → ∀𝑧(𝑧𝐷 → ¬ 𝑧𝑅𝑦))
22 con2b 350 . . . . . . . . . . . . . . . . 17 ((𝑧𝐷 → ¬ 𝑧𝑅𝑦) ↔ (𝑧𝑅𝑦 → ¬ 𝑧𝐷))
2322albii 1914 . . . . . . . . . . . . . . . 16 (∀𝑧(𝑧𝐷 → ¬ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷))
2421, 23sylib 209 . . . . . . . . . . . . . . 15 (𝜃 → ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷))
25 bnj1418 31556 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧𝑅𝑦)
2625imim1i 63 . . . . . . . . . . . . . . . 16 ((𝑧𝑅𝑦 → ¬ 𝑧𝐷) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2726alimi 1906 . . . . . . . . . . . . . . 15 (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2824, 27syl 17 . . . . . . . . . . . . . 14 (𝜃 → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2928bnj1142 31308 . . . . . . . . . . . . 13 (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅) ¬ 𝑧𝐷)
30 bnj1523.8 . . . . . . . . . . . . . 14 𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}
315bnj1309 31538 . . . . . . . . . . . . . . . . . . 19 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
327, 31bnj1307 31539 . . . . . . . . . . . . . . . . . 18 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3332nfcii 2898 . . . . . . . . . . . . . . . . 17 𝑥𝐶
3433nfuni 4600 . . . . . . . . . . . . . . . 16 𝑥 𝐶
358, 34nfcxfr 2905 . . . . . . . . . . . . . . 15 𝑥𝐹
3635nfcrii 2900 . . . . . . . . . . . . . 14 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
3730, 36bnj1534 31371 . . . . . . . . . . . . 13 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
3829, 18, 37bnj1533 31370 . . . . . . . . . . . 12 (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅)(𝐹𝑧) = (𝐻𝑧))
3913, 17, 19, 38bnj1536 31372 . . . . . . . . . . 11 (𝜃 → (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) = (𝐻 ↾ pred(𝑦, 𝐴, 𝑅)))
4039opeq2d 4566 . . . . . . . . . 10 (𝜃 → ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
4140fveq2d 6379 . . . . . . . . 9 (𝜃 → (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
425, 6, 7, 8bnj1500 31584 . . . . . . . . . . . . . . 15 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
431, 42bnj835 31277 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
442, 43bnj832 31276 . . . . . . . . . . . . 13 (𝜓 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
454, 44bnj835 31277 . . . . . . . . . . . 12 (𝜒 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
4645, 36bnj1529 31586 . . . . . . . . . . 11 (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
473, 46bnj835 31277 . . . . . . . . . 10 (𝜃 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
4830ssrab3 3848 . . . . . . . . . . 11 𝐷𝐴
493simp2bi 1176 . . . . . . . . . . 11 (𝜃𝑦𝐷)
5048, 49bnj1213 31317 . . . . . . . . . 10 (𝜃𝑦𝐴)
5147, 50bnj1294 31336 . . . . . . . . 9 (𝜃 → (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
521simp3bi 1177 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
532, 52bnj832 31276 . . . . . . . . . . . . 13 (𝜓 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
544, 53bnj835 31277 . . . . . . . . . . . 12 (𝜒 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55 ax-5 2005 . . . . . . . . . . . 12 (𝑣𝐻 → ∀𝑥 𝑣𝐻)
5654, 55bnj1529 31586 . . . . . . . . . . 11 (𝜒 → ∀𝑦𝐴 (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
573, 56bnj835 31277 . . . . . . . . . 10 (𝜃 → ∀𝑦𝐴 (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
5857, 50bnj1294 31336 . . . . . . . . 9 (𝜃 → (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
5941, 51, 583eqtr4d 2809 . . . . . . . 8 (𝜃 → (𝐹𝑦) = (𝐻𝑦))
6030, 36bnj1534 31371 . . . . . . . . . . 11 𝐷 = {𝑦𝐴 ∣ (𝐹𝑦) ≠ (𝐻𝑦)}
6160bnj1538 31373 . . . . . . . . . 10 (𝑦𝐷 → (𝐹𝑦) ≠ (𝐻𝑦))
623, 61bnj836 31278 . . . . . . . . 9 (𝜃 → (𝐹𝑦) ≠ (𝐻𝑦))
6362neneqd 2942 . . . . . . . 8 (𝜃 → ¬ (𝐹𝑦) = (𝐻𝑦))
6459, 63pm2.65i 185 . . . . . . 7 ¬ 𝜃
6564nex 1895 . . . . . 6 ¬ ∃𝑦𝜃
661simp1bi 1175 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
672, 66bnj832 31276 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
684, 67bnj835 31277 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
6948a1i 11 . . . . . . . 8 (𝜒𝐷𝐴)
704simp2bi 1176 . . . . . . . . . 10 (𝜒𝑥𝐴)
714simp3bi 1177 . . . . . . . . . 10 (𝜒 → (𝐹𝑥) ≠ (𝐻𝑥))
7230rabeq2i 3346 . . . . . . . . . 10 (𝑥𝐷 ↔ (𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))
7370, 71, 72sylanbrc 578 . . . . . . . . 9 (𝜒𝑥𝐷)
7473ne0d 4086 . . . . . . . 8 (𝜒𝐷 ≠ ∅)
75 bnj69 31526 . . . . . . . 8 ((𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅) → ∃𝑦𝐷𝑧𝐷 ¬ 𝑧𝑅𝑦)
7668, 69, 74, 75syl3anc 1490 . . . . . . 7 (𝜒 → ∃𝑦𝐷𝑧𝐷 ¬ 𝑧𝑅𝑦)
7776, 3bnj1209 31315 . . . . . 6 (𝜒 → ∃𝑦𝜃)
7865, 77mto 188 . . . . 5 ¬ 𝜒
7978nex 1895 . . . 4 ¬ ∃𝑥𝜒
802simprbi 490 . . . . . 6 (𝜓𝐹𝐻)
8111, 15, 80, 36bnj1542 31375 . . . . 5 (𝜓 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐻𝑥))
825, 6, 7, 8, 1, 2bnj1525 31585 . . . . 5 (𝜓 → ∀𝑥𝜓)
8381, 4, 82bnj1521 31369 . . . 4 (𝜓 → ∃𝑥𝜒)
8479, 83mto 188 . . 3 ¬ 𝜓
852, 84bnj1541 31374 . 2 (𝜑𝐹 = 𝐻)
861, 85sylbir 226 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 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  wss 3732  c0 4079  cop 4340   cuni 4594   class class class wbr 4809  cres 5279   Fn wfn 6063  cfv 6068   predc-bnj14 31205   FrSe w-bnj15 31209
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 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-bnj17 31204  df-bnj14 31206  df-bnj13 31208  df-bnj15 31210  df-bnj18 31212  df-bnj19 31214
This theorem is referenced by:  bnj1522  31588
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