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Theorem bnj1311 34333
Description: Technical lemma for bnj60 34371. 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 260 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
21bnj1232 34112 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑅 FrSe 𝐴)
3 ssrab2 4076 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐷
4 bnj1311.4 . . . . . . . . 9 𝐷 = (dom 𝑔 ∩ dom )
51bnj1235 34113 . . . . . . . . . . 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 2730 . . . . . . . . . . . 12 𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9 eqid 2730 . . . . . . . . . . . 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 34322 . . . . . . . . . . 11 𝐶 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
115, 10eleqtrdi 2841 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))})
12 abid 2711 . . . . . . . . . . . . . 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 34115 . . . . . . . . . . . . 13 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑𝐵 𝑔 Fn 𝑑)
1413bnj1196 34103 . . . . . . . . . . . 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 2875 . . . . . . . . . . . . . 14 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1716simplbi 496 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
18 fndm 6651 . . . . . . . . . . . . 13 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1917, 18bnj1241 34116 . . . . . . . . . . . 12 ((𝑑𝐵𝑔 Fn 𝑑) → dom 𝑔𝐴)
2014, 19bnj593 34054 . . . . . . . . . . 11 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑dom 𝑔𝐴)
2120bnj937 34080 . . . . . . . . . 10 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔𝐴)
22 ssinss1 4236 . . . . . . . . . 10 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2311, 21, 223syl 18 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
244, 23eqsstrid 4029 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝐷𝐴)
253, 24sstrid 3992 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴)
26 eqid 2730 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
27 biid 260 . . . . . . . 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 34326 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅)
29 nfrab1 3449 . . . . . . . . 9 𝑥{𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
3029nfcrii 2893 . . . . . . . 8 (𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → ∀𝑥 𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
3130bnj1228 34320 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
322, 25, 28, 31syl3anc 1369 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
33 ax-5 1911 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
3415bnj1309 34331 . . . . . . . . 9 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
357, 34bnj1307 34332 . . . . . . . 8 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3635hblem 2863 . . . . . . 7 (𝑔𝐶 → ∀𝑥 𝑔𝐶)
3735hblem 2863 . . . . . . 7 (𝐶 → ∀𝑥 𝐶)
38 ax-5 1911 . . . . . . 7 ((𝑔𝐷) ≠ (𝐷) → ∀𝑥(𝑔𝐷) ≠ (𝐷))
3933, 36, 37, 38bnj982 34087 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∀𝑥(𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
4032, 27, 39bnj1521 34160 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥))
41 simp2 1135 . . . . 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 34327 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}) = ∅)
43423adant1 1128 . . . . . . . 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 34329 . . . . . . 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 2730 . . . . . . 7 𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
46 eqid 2730 . . . . . . 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 34330 . . . . . 6 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔𝑥) = (𝑥))
4826bnj1538 34164 . . . . . . 7 (𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → (𝑔𝑥) ≠ (𝑥))
4948necon2bi 2969 . . . . . 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 34128 . . . 4 ¬ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷))
52 df-bnj17 33996 . . . 4 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷)))
5351, 52mtbi 321 . . 3 ¬ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷))
5453imnani 399 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → ¬ (𝑔𝐷) ≠ (𝐷))
55 nne 2942 . 2 (¬ (𝑔𝐷) ≠ (𝐷) ↔ (𝑔𝐷) = (𝐷))
5654, 55sylib 217 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  {cab 2707  wne 2938  wral 3059  wrex 3068  {crab 3430  cin 3946  wss 3947  c0 4321  cop 4633   class class class wbr 5147  dom cdm 5675  cres 5677   Fn wfn 6537  cfv 6542  w-bnj17 33995   predc-bnj14 33997   FrSe w-bnj15 34001
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7858  df-1o 8468  df-bnj17 33996  df-bnj14 33998  df-bnj13 34000  df-bnj15 34002  df-bnj18 34004  df-bnj19 34006
This theorem is referenced by:  bnj1326  34335  bnj60  34371
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