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 31409
Description: Technical lemma for bnj60 31447. 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 252 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
21bnj1232 31191 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑅 FrSe 𝐴)
3 ssrab2 3878 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐷
4 bnj1311.4 . . . . . . . . 9 𝐷 = (dom 𝑔 ∩ dom )
51bnj1235 31192 . . . . . . . . . . 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 2802 . . . . . . . . . . . 12 𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9 eqid 2802 . . . . . . . . . . . 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 31398 . . . . . . . . . . 11 𝐶 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
115, 10syl6eleq 2891 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))})
12 abid 2790 . . . . . . . . . . . . . 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 31194 . . . . . . . . . . . . 13 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑𝐵 𝑔 Fn 𝑑)
1413bnj1196 31182 . . . . . . . . . . . 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 2915 . . . . . . . . . . . . . 14 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1716simplbi 487 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
18 fndm 6195 . . . . . . . . . . . . 13 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1917, 18bnj1241 31195 . . . . . . . . . . . 12 ((𝑑𝐵𝑔 Fn 𝑑) → dom 𝑔𝐴)
2014, 19bnj593 31132 . . . . . . . . . . 11 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑dom 𝑔𝐴)
2120bnj937 31159 . . . . . . . . . 10 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔𝐴)
22 ssinss1 4032 . . . . . . . . . 10 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2311, 21, 223syl 18 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
244, 23syl5eqss 3840 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝐷𝐴)
253, 24syl5ss 3803 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴)
26 eqid 2802 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
27 biid 252 . . . . . . . 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 31402 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅)
29 nfrab1 3307 . . . . . . . . 9 𝑥{𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
3029nfcrii 2937 . . . . . . . 8 (𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → ∀𝑥 𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
3130bnj1228 31396 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
322, 25, 28, 31syl3anc 1483 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
33 ax-5 2001 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
3415bnj1309 31407 . . . . . . . . 9 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
357, 34bnj1307 31408 . . . . . . . 8 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3635hblem 2911 . . . . . . 7 (𝑔𝐶 → ∀𝑥 𝑔𝐶)
3735hblem 2911 . . . . . . 7 (𝐶 → ∀𝑥 𝐶)
38 ax-5 2001 . . . . . . 7 ((𝑔𝐷) ≠ (𝐷) → ∀𝑥(𝑔𝐷) ≠ (𝐷))
3933, 36, 37, 38bnj982 31166 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∀𝑥(𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
4032, 27, 39bnj1521 31238 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥))
41 simp2 1160 . . . . 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 31403 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}) = ∅)
43423adant1 1153 . . . . . . . 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 31405 . . . . . . 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 2802 . . . . . . 7 𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
46 eqid 2802 . . . . . . 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 31406 . . . . . 6 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔𝑥) = (𝑥))
4826bnj1538 31242 . . . . . . 7 (𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → (𝑔𝑥) ≠ (𝑥))
4948necon2bi 3004 . . . . . 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 31207 . . . 4 ¬ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷))
52 df-bnj17 31072 . . . 4 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷)))
5351, 52mtbi 313 . . 3 ¬ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷))
5453imnani 389 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → ¬ (𝑔𝐷) ≠ (𝐷))
55 nne 2978 . 2 (¬ (𝑔𝐷) ≠ (𝐷) ↔ (𝑔𝐷) = (𝐷))
5654, 55sylib 209 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2155  {cab 2788  wne 2974  wral 3092  wrex 3093  {crab 3096  cin 3762  wss 3763  c0 4110  cop 4370   class class class wbr 4837  dom cdm 5305  cres 5307   Fn wfn 6090  cfv 6095  w-bnj17 31071   predc-bnj14 31073   FrSe w-bnj15 31077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-reg 8730  ax-inf2 8779
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-om 7290  df-1o 7790  df-bnj17 31072  df-bnj14 31074  df-bnj13 31076  df-bnj15 31078  df-bnj18 31080  df-bnj19 31082
This theorem is referenced by:  bnj1326  31411  bnj60  31447
  Copyright terms: Public domain W3C validator