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Theorem bnj1311 35038
Description: Technical lemma for bnj60 35076. 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
bnj1311.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1311.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1311.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1311.4 𝐷 = (dom 𝑔 ∩ dom )
Assertion
Ref Expression
bnj1311 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓,𝑔   𝐵,,𝑓   𝐷,𝑑,𝑥   𝐺,𝑑,𝑓,𝑔   ,𝐺,𝑑   𝑅,𝑑,𝑓,𝑥   𝑔,𝑌   ,𝑌   𝑥,𝑔   𝑥,
Allowed substitution hints:   𝐴(𝑔,)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑔,,𝑑)   𝐷(𝑓,𝑔,)   𝑅(𝑔,)   𝐺(𝑥)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1311
Dummy variables 𝑤 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 261 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
21bnj1232 34817 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑅 FrSe 𝐴)
3 ssrab2 4080 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐷
4 bnj1311.4 . . . . . . . . 9 𝐷 = (dom 𝑔 ∩ dom )
51bnj1235 34818 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑔𝐶)
6 bnj1311.2 . . . . . . . . . . . 12 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
7 bnj1311.3 . . . . . . . . . . . 12 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
8 eqid 2737 . . . . . . . . . . . 12 𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9 eqid 2737 . . . . . . . . . . . 12 {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
106, 7, 8, 9bnj1234 35027 . . . . . . . . . . 11 𝐶 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
115, 10eleqtrdi 2851 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))})
12 abid 2718 . . . . . . . . . . . . . 14 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} ↔ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
1312bnj1238 34820 . . . . . . . . . . . . 13 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑𝐵 𝑔 Fn 𝑑)
1413bnj1196 34808 . . . . . . . . . . . 12 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑(𝑑𝐵𝑔 Fn 𝑑))
15 bnj1311.1 . . . . . . . . . . . . . . 15 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
1615eqabri 2885 . . . . . . . . . . . . . 14 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1716simplbi 497 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
18 fndm 6671 . . . . . . . . . . . . 13 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1917, 18bnj1241 34821 . . . . . . . . . . . 12 ((𝑑𝐵𝑔 Fn 𝑑) → dom 𝑔𝐴)
2014, 19bnj593 34759 . . . . . . . . . . 11 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑dom 𝑔𝐴)
2120bnj937 34785 . . . . . . . . . 10 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔𝐴)
22 ssinss1 4246 . . . . . . . . . 10 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2311, 21, 223syl 18 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
244, 23eqsstrid 4022 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝐷𝐴)
253, 24sstrid 3995 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴)
26 eqid 2737 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
27 biid 261 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥))
2815, 6, 7, 4, 26, 1, 27bnj1253 35031 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅)
29 nfrab1 3457 . . . . . . . . 9 𝑥{𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
3029nfcrii 2900 . . . . . . . 8 (𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → ∀𝑥 𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
3130bnj1228 35025 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
322, 25, 28, 31syl3anc 1373 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
33 ax-5 1910 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
3415bnj1309 35036 . . . . . . . . 9 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
357, 34bnj1307 35037 . . . . . . . 8 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3635hblem 2873 . . . . . . 7 (𝑔𝐶 → ∀𝑥 𝑔𝐶)
3735hblem 2873 . . . . . . 7 (𝐶 → ∀𝑥 𝐶)
38 ax-5 1910 . . . . . . 7 ((𝑔𝐷) ≠ (𝐷) → ∀𝑥(𝑔𝐷) ≠ (𝐷))
3933, 36, 37, 38bnj982 34792 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∀𝑥(𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
4032, 27, 39bnj1521 34865 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥))
41 simp2 1138 . . . . 5 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
4215, 6, 7, 4, 26, 1, 27bnj1279 35032 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}) = ∅)
43423adant1 1131 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}) = ∅)
4415, 6, 7, 4, 26, 1, 27, 43bnj1280 35034 . . . . . . 7 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
45 eqid 2737 . . . . . . 7 𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
46 eqid 2737 . . . . . . 7 { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺‘⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
4715, 6, 7, 4, 26, 1, 27, 44, 8, 9, 45, 46bnj1296 35035 . . . . . 6 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔𝑥) = (𝑥))
4826bnj1538 34869 . . . . . . 7 (𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → (𝑔𝑥) ≠ (𝑥))
4948necon2bi 2971 . . . . . 6 ((𝑔𝑥) = (𝑥) → ¬ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
5047, 49syl 17 . . . . 5 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → ¬ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
5140, 41, 50bnj1304 34833 . . . 4 ¬ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷))
52 df-bnj17 34701 . . . 4 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷)))
5351, 52mtbi 322 . . 3 ¬ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷))
5453imnani 400 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → ¬ (𝑔𝐷) ≠ (𝐷))
55 nne 2944 . 2 (¬ (𝑔𝐷) ≠ (𝐷) ↔ (𝑔𝐷) = (𝐷))
5654, 55sylib 218 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  cin 3950  wss 3951  c0 4333  cop 4632   class class class wbr 5143  dom cdm 5685  cres 5687   Fn wfn 6556  cfv 6561  w-bnj17 34700   predc-bnj14 34702   FrSe w-bnj15 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-bnj17 34701  df-bnj14 34703  df-bnj13 34705  df-bnj15 34707  df-bnj18 34709  df-bnj19 34711
This theorem is referenced by:  bnj1326  35040  bnj60  35076
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