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Theorem bnj1311 33636
Description: Technical lemma for bnj60 33674. 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 33415 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑅 FrSe 𝐴)
3 ssrab2 4037 . . . . . . . 8 {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐷
4 bnj1311.4 . . . . . . . . 9 𝐷 = (dom 𝑔 ∩ dom )
51bnj1235 33416 . . . . . . . . . . 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 2736 . . . . . . . . . . . 12 𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
9 eqid 2736 . . . . . . . . . . . 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 33625 . . . . . . . . . . 11 𝐶 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
115, 10eleqtrdi 2848 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))})
12 abid 2717 . . . . . . . . . . . . . 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 33418 . . . . . . . . . . . . 13 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑𝐵 𝑔 Fn 𝑑)
1413bnj1196 33406 . . . . . . . . . . . 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)}
1615eqabi 2881 . . . . . . . . . . . . . 14 (𝑑𝐵 ↔ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
1716simplbi 498 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
18 fndm 6605 . . . . . . . . . . . . 13 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1917, 18bnj1241 33419 . . . . . . . . . . . 12 ((𝑑𝐵𝑔 Fn 𝑑) → dom 𝑔𝐴)
2014, 19bnj593 33357 . . . . . . . . . . 11 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑dom 𝑔𝐴)
2120bnj937 33383 . . . . . . . . . 10 (𝑔 ∈ {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → dom 𝑔𝐴)
22 ssinss1 4197 . . . . . . . . . 10 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2311, 21, 223syl 18 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
244, 23eqsstrid 3992 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → 𝐷𝐴)
253, 24sstrid 3955 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴)
26 eqid 2736 . . . . . . . 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 33629 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅)
29 nfrab1 3426 . . . . . . . . 9 𝑥{𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
3029nfcrii 2899 . . . . . . . 8 (𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → ∀𝑥 𝑧 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)})
3130bnj1228 33623 . . . . . . 7 ((𝑅 FrSe 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ⊆ 𝐴 ∧ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
322, 25, 28, 31syl3anc 1371 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥)
33 ax-5 1913 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥 𝑅 FrSe 𝐴)
3415bnj1309 33634 . . . . . . . . 9 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
357, 34bnj1307 33635 . . . . . . . 8 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
3635hblem 2869 . . . . . . 7 (𝑔𝐶 → ∀𝑥 𝑔𝐶)
3735hblem 2869 . . . . . . 7 (𝐶 → ∀𝑥 𝐶)
38 ax-5 1913 . . . . . . 7 ((𝑔𝐷) ≠ (𝐷) → ∀𝑥(𝑔𝐷) ≠ (𝐷))
3933, 36, 37, 38bnj982 33390 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∀𝑥(𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
4032, 27, 39bnj1521 33463 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) → ∃𝑥((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥))
41 simp2 1137 . . . . 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 33630 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}) = ∅)
43423adant1 1130 . . . . . . . 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 33632 . . . . . . 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 2736 . . . . . . 7 𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
46 eqid 2736 . . . . . . 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 33633 . . . . . 6 (((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ∧ 𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ∧ ∀𝑦 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ¬ 𝑦𝑅𝑥) → (𝑔𝑥) = (𝑥))
4826bnj1538 33467 . . . . . . 7 (𝑥 ∈ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} → (𝑔𝑥) ≠ (𝑥))
4948necon2bi 2974 . . . . . 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 33431 . . . 4 ¬ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷))
52 df-bnj17 33299 . . . 4 ((𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷)))
5351, 52mtbi 321 . . 3 ¬ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) ∧ (𝑔𝐷) ≠ (𝐷))
5453imnani 401 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → ¬ (𝑔𝐷) ≠ (𝐷))
55 nne 2947 . 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 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  {crab 3407  cin 3909  wss 3910  c0 4282  cop 4592   class class class wbr 5105  dom cdm 5633  cres 5635   Fn wfn 6491  cfv 6496  w-bnj17 33298   predc-bnj14 33300   FrSe w-bnj15 33304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-bnj17 33299  df-bnj14 33301  df-bnj13 33303  df-bnj15 33305  df-bnj18 33307  df-bnj19 33309
This theorem is referenced by:  bnj1326  33638  bnj60  33674
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