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Theorem bnj1523 34638
Description: Technical lemma for bnj1522 34639. 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 34629 . . . . . . . . . . . . . . . 16 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
101, 9bnj835 34326 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝐴)
112, 10bnj832 34325 . . . . . . . . . . . . . 14 (𝜓𝐹 Fn 𝐴)
124, 11bnj835 34326 . . . . . . . . . . . . 13 (𝜒𝐹 Fn 𝐴)
133, 12bnj835 34326 . . . . . . . . . . . 12 (𝜃𝐹 Fn 𝐴)
141simp2bi 1144 . . . . . . . . . . . . . . 15 (𝜑𝐻 Fn 𝐴)
152, 14bnj832 34325 . . . . . . . . . . . . . 14 (𝜓𝐻 Fn 𝐴)
164, 15bnj835 34326 . . . . . . . . . . . . 13 (𝜒𝐻 Fn 𝐴)
173, 16bnj835 34326 . . . . . . . . . . . 12 (𝜃𝐻 Fn 𝐴)
18 bnj213 34449 . . . . . . . . . . . . 13 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
1918a1i 11 . . . . . . . . . . . 12 (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
203simp3bi 1145 . . . . . . . . . . . . . . . . 17 (𝜃 → ∀𝑧𝐷 ¬ 𝑧𝑅𝑦)
2120bnj1211 34364 . . . . . . . . . . . . . . . 16 (𝜃 → ∀𝑧(𝑧𝐷 → ¬ 𝑧𝑅𝑦))
22 con2b 359 . . . . . . . . . . . . . . . . 17 ((𝑧𝐷 → ¬ 𝑧𝑅𝑦) ↔ (𝑧𝑅𝑦 → ¬ 𝑧𝐷))
2322albii 1814 . . . . . . . . . . . . . . . 16 (∀𝑧(𝑧𝐷 → ¬ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷))
2421, 23sylib 217 . . . . . . . . . . . . . . 15 (𝜃 → ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷))
25 bnj1418 34607 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧𝑅𝑦)
2625imim1i 63 . . . . . . . . . . . . . . . 16 ((𝑧𝑅𝑦 → ¬ 𝑧𝐷) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2726alimi 1806 . . . . . . . . . . . . . . 15 (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝐷) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2824, 27syl 17 . . . . . . . . . . . . . 14 (𝜃 → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧𝐷))
2928bnj1142 34356 . . . . . . . . . . . . 13 (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅) ¬ 𝑧𝐷)
30 bnj1523.8 . . . . . . . . . . . . . 14 𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}
315bnj1309 34589 . . . . . . . . . . . . . . . . . . 19 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
327, 31bnj1307 34590 . . . . . . . . . . . . . . . . . 18 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3332nfcii 2882 . . . . . . . . . . . . . . . . 17 𝑥𝐶
3433nfuni 4910 . . . . . . . . . . . . . . . 16 𝑥 𝐶
358, 34nfcxfr 2896 . . . . . . . . . . . . . . 15 𝑥𝐹
3635nfcrii 2890 . . . . . . . . . . . . . 14 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
3730, 36bnj1534 34420 . . . . . . . . . . . . 13 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
3829, 18, 37bnj1533 34419 . . . . . . . . . . . 12 (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅)(𝐹𝑧) = (𝐻𝑧))
3913, 17, 19, 38bnj1536 34421 . . . . . . . . . . 11 (𝜃 → (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) = (𝐻 ↾ pred(𝑦, 𝐴, 𝑅)))
4039opeq2d 4876 . . . . . . . . . 10 (𝜃 → ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
4140fveq2d 6895 . . . . . . . . 9 (𝜃 → (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
425, 6, 7, 8bnj1500 34635 . . . . . . . . . . . . . . 15 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
431, 42bnj835 34326 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
442, 43bnj832 34325 . . . . . . . . . . . . 13 (𝜓 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
454, 44bnj835 34326 . . . . . . . . . . . 12 (𝜒 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
4645, 36bnj1529 34637 . . . . . . . . . . 11 (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
473, 46bnj835 34326 . . . . . . . . . 10 (𝜃 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
4830ssrab3 4076 . . . . . . . . . . 11 𝐷𝐴
493simp2bi 1144 . . . . . . . . . . 11 (𝜃𝑦𝐷)
5048, 49bnj1213 34365 . . . . . . . . . 10 (𝜃𝑦𝐴)
5147, 50bnj1294 34384 . . . . . . . . 9 (𝜃 → (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
521simp3bi 1145 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
532, 52bnj832 34325 . . . . . . . . . . . . 13 (𝜓 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
544, 53bnj835 34326 . . . . . . . . . . . 12 (𝜒 → ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55 ax-5 1906 . . . . . . . . . . . 12 (𝑣𝐻 → ∀𝑥 𝑣𝐻)
5654, 55bnj1529 34637 . . . . . . . . . . 11 (𝜒 → ∀𝑦𝐴 (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
573, 56bnj835 34326 . . . . . . . . . 10 (𝜃 → ∀𝑦𝐴 (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
5857, 50bnj1294 34384 . . . . . . . . 9 (𝜃 → (𝐻𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
5941, 51, 583eqtr4d 2777 . . . . . . . 8 (𝜃 → (𝐹𝑦) = (𝐻𝑦))
6030, 36bnj1534 34420 . . . . . . . . . . 11 𝐷 = {𝑦𝐴 ∣ (𝐹𝑦) ≠ (𝐻𝑦)}
6160bnj1538 34422 . . . . . . . . . 10 (𝑦𝐷 → (𝐹𝑦) ≠ (𝐻𝑦))
623, 61bnj836 34327 . . . . . . . . 9 (𝜃 → (𝐹𝑦) ≠ (𝐻𝑦))
6362neneqd 2940 . . . . . . . 8 (𝜃 → ¬ (𝐹𝑦) = (𝐻𝑦))
6459, 63pm2.65i 193 . . . . . . 7 ¬ 𝜃
6564nex 1795 . . . . . 6 ¬ ∃𝑦𝜃
661simp1bi 1143 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
672, 66bnj832 34325 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
684, 67bnj835 34326 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
6948a1i 11 . . . . . . . 8 (𝜒𝐷𝐴)
704simp2bi 1144 . . . . . . . . . 10 (𝜒𝑥𝐴)
714simp3bi 1145 . . . . . . . . . 10 (𝜒 → (𝐹𝑥) ≠ (𝐻𝑥))
7230reqabi 3449 . . . . . . . . . 10 (𝑥𝐷 ↔ (𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))
7370, 71, 72sylanbrc 582 . . . . . . . . 9 (𝜒𝑥𝐷)
7473ne0d 4331 . . . . . . . 8 (𝜒𝐷 ≠ ∅)
75 bnj69 34577 . . . . . . . 8 ((𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅) → ∃𝑦𝐷𝑧𝐷 ¬ 𝑧𝑅𝑦)
7668, 69, 74, 75syl3anc 1369 . . . . . . 7 (𝜒 → ∃𝑦𝐷𝑧𝐷 ¬ 𝑧𝑅𝑦)
7776, 3bnj1209 34363 . . . . . 6 (𝜒 → ∃𝑦𝜃)
7865, 77mto 196 . . . . 5 ¬ 𝜒
7978nex 1795 . . . 4 ¬ ∃𝑥𝜒
802simprbi 496 . . . . . 6 (𝜓𝐹𝐻)
8111, 15, 80, 36bnj1542 34424 . . . . 5 (𝜓 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐻𝑥))
825, 6, 7, 8, 1, 2bnj1525 34636 . . . . 5 (𝜓 → ∀𝑥𝜓)
8381, 4, 82bnj1521 34418 . . . 4 (𝜓 → ∃𝑥𝜒)
8479, 83mto 196 . . 3 ¬ 𝜓
852, 84bnj1541 34423 . 2 (𝜑𝐹 = 𝐻)
861, 85sylbir 234 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 205  wa 395  w3a 1085  wal 1532   = wceq 1534  wex 1774  wcel 2099  {cab 2704  wne 2935  wral 3056  wrex 3065  {crab 3427  wss 3944  c0 4318  cop 4630   cuni 4903   class class class wbr 5142  cres 5674   Fn wfn 6537  cfv 6542   predc-bnj14 34255   FrSe w-bnj15 34259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607  ax-inf2 9656
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7865  df-1o 8480  df-bnj17 34254  df-bnj14 34256  df-bnj13 34258  df-bnj15 34260  df-bnj18 34262  df-bnj19 34264
This theorem is referenced by:  bnj1522  34639
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