Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1311 Structured version   Visualization version   GIF version

Theorem bnj1311 30140
Description: Technical lemma for bnj60 30178. 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 250 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
21bnj1232 29922 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑅 FrSe 𝐴)
3 ssrab2 3650 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐷
4 bnj1311.4 . . . . . . . . 9 𝐷 = (dom 𝑔 ∩ dom )
51bnj1235 29923 . . . . . . . . . . 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 2610 . . . . . . . . . . . 12 𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9 eqid 2610 . . . . . . . . . . . 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 30129 . . . . . . . . . . 11 𝐶 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
115, 10syl6eleq 2698 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))})
12 abid 2598 . . . . . . . . . . . . . 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 29925 . . . . . . . . . . . . 13 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑𝐵 𝑔 Fn 𝑑)
1413bnj1196 29913 . . . . . . . . . . . 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)}
1615abeq2i 2722 . . . . . . . . . . . . . 14 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1716simplbi 475 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
18 fndm 5890 . . . . . . . . . . . . 13 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1917, 18bnj1241 29926 . . . . . . . . . . . 12 ((𝑑𝐵𝑔 Fn 𝑑) → dom 𝑔𝐴)
2014, 19bnj593 29863 . . . . . . . . . . 11 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑dom 𝑔𝐴)
2120bnj937 29890 . . . . . . . . . 10 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔𝐴)
22 ssinss1 3803 . . . . . . . . . 10 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2311, 21, 223syl 18 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
244, 23syl5eqss 3612 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝐷𝐴)
253, 24syl5ss 3579 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴)
26 eqid 2610 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
27 biid 250 . . . . . . . 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 30133 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅)
29 nfrab1 3099 . . . . . . . . 9 𝑥{𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
3029nfcrii 2744 . . . . . . . 8 (𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → ∀𝑥 𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
3130bnj1228 30127 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
322, 25, 28, 31syl3anc 1318 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
33 ax-5 1827 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
3415bnj1309 30138 . . . . . . . . 9 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
357, 34bnj1307 30139 . . . . . . . 8 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3635hblem 2718 . . . . . . 7 (𝑔𝐶 → ∀𝑥 𝑔𝐶)
3735hblem 2718 . . . . . . 7 (𝐶 → ∀𝑥 𝐶)
38 ax-5 1827 . . . . . . 7 ((𝑔𝐷) ≠ (𝐷) → ∀𝑥(𝑔𝐷) ≠ (𝐷))
3933, 36, 37, 38bnj982 29897 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∀𝑥(𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
4032, 27, 39bnj1521 29969 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥))
41 simp2 1055 . . . . 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 30134 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}) = ∅)
43423adant1 1072 . . . . . . . 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 30136 . . . . . . 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 2610 . . . . . . 7 𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
46 eqid 2610 . . . . . . 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 30137 . . . . . 6 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔𝑥) = (𝑥))
4826bnj1538 29973 . . . . . . 7 (𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → (𝑔𝑥) ≠ (𝑥))
4948necon2bi 2812 . . . . . 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 29938 . . . 4 ¬ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷))
52 df-bnj17 29800 . . . 4 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷)))
5351, 52mtbi 311 . . 3 ¬ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷))
5453imnani 438 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → ¬ (𝑔𝐷) ≠ (𝐷))
55 nne 2786 . 2 (¬ (𝑔𝐷) ≠ (𝐷) ↔ (𝑔𝐷) = (𝐷))
5654, 55sylib 207 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  {cab 2596  wne 2780  wral 2896  wrex 2897  {crab 2900  cin 3539  wss 3540  c0 3874  cop 4131   class class class wbr 4578  dom cdm 5028  cres 5030   Fn wfn 5785  cfv 5790  w-bnj17 29799   predc-bnj14 29801   FrSe w-bnj15 29805
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-reg 8358  ax-inf2 8399
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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6936  df-1o 7425  df-bnj17 29800  df-bnj14 29802  df-bnj13 29804  df-bnj15 29806  df-bnj18 29808  df-bnj19 29810
This theorem is referenced by:  bnj1326  30142  bnj60  30178
  Copyright terms: Public domain W3C validator