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Theorem bnj1280 30342
 Description: Technical lemma for bnj60 30384. 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
bnj1280.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1280.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1280.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1280.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1280.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1280.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1280.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
bnj1280.17 (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
Assertion
Ref Expression
bnj1280 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑑,𝑓   𝐵,𝑓,𝑔   𝐵,,𝑓   𝐷,𝑑,𝑥   𝑓,𝐺,𝑔   ,𝐺   𝑅,𝑑,𝑓   𝑔,𝑌   ,𝑌   𝑔,𝑑   𝑥,𝑓,𝑔   ,𝑑,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑔,)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑦,𝑓,𝑔,)   𝑅(𝑥,𝑦,𝑔,)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1280
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj1280.1 . . . . . . . 8 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj1280.2 . . . . . . . 8 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj1280.3 . . . . . . . 8 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 bnj1280.4 . . . . . . . 8 𝐷 = (dom 𝑔 ∩ dom )
5 bnj1280.5 . . . . . . . 8 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
6 bnj1280.6 . . . . . . . 8 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
7 bnj1280.7 . . . . . . . 8 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
81, 2, 3, 4, 5, 6, 7bnj1286 30341 . . . . . . 7 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
98sseld 3567 . . . . . 6 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝐷))
10 bnj1280.17 . . . . . . . . 9 (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
11 disj1 3971 . . . . . . . . 9 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅ ↔ ∀𝑧(𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧𝐸))
1210, 11sylib 207 . . . . . . . 8 (𝜓 → ∀𝑧(𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧𝐸))
131219.21bi 2047 . . . . . . 7 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧𝐸))
14 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑔𝑥) = (𝑔𝑧))
15 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥) = (𝑧))
1614, 15neeq12d 2843 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑔𝑥) ≠ (𝑥) ↔ (𝑔𝑧) ≠ (𝑧)))
1716, 5elrab2 3333 . . . . . . . . 9 (𝑧𝐸 ↔ (𝑧𝐷 ∧ (𝑔𝑧) ≠ (𝑧)))
1817notbii 309 . . . . . . . 8 𝑧𝐸 ↔ ¬ (𝑧𝐷 ∧ (𝑔𝑧) ≠ (𝑧)))
19 imnan 437 . . . . . . . 8 ((𝑧𝐷 → ¬ (𝑔𝑧) ≠ (𝑧)) ↔ ¬ (𝑧𝐷 ∧ (𝑔𝑧) ≠ (𝑧)))
20 nne 2786 . . . . . . . . 9 (¬ (𝑔𝑧) ≠ (𝑧) ↔ (𝑔𝑧) = (𝑧))
2120imbi2i 325 . . . . . . . 8 ((𝑧𝐷 → ¬ (𝑔𝑧) ≠ (𝑧)) ↔ (𝑧𝐷 → (𝑔𝑧) = (𝑧)))
2218, 19, 213bitr2i 287 . . . . . . 7 𝑧𝐸 ↔ (𝑧𝐷 → (𝑔𝑧) = (𝑧)))
2313, 22syl6ib 240 . . . . . 6 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑧𝐷 → (𝑔𝑧) = (𝑧))))
249, 23mpdd 42 . . . . 5 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑔𝑧) = (𝑧)))
2524imp 444 . . . 4 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑔𝑧) = (𝑧))
26 fvres 6117 . . . . . 6 (𝑧𝐷 → ((𝑔𝐷)‘𝑧) = (𝑔𝑧))
279, 26syl6 34 . . . . 5 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ((𝑔𝐷)‘𝑧) = (𝑔𝑧)))
2827imp 444 . . . 4 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔𝐷)‘𝑧) = (𝑔𝑧))
29 fvres 6117 . . . . . 6 (𝑧𝐷 → ((𝐷)‘𝑧) = (𝑧))
309, 29syl6 34 . . . . 5 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ((𝐷)‘𝑧) = (𝑧)))
3130imp 444 . . . 4 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝐷)‘𝑧) = (𝑧))
3225, 28, 313eqtr4d 2654 . . 3 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧))
3332ralrimiva 2949 . 2 (𝜓 → ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧))
348resabs1d 5348 . . . 4 (𝜓 → ((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
358resabs1d 5348 . . . 4 (𝜓 → ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
3634, 35eqeq12d 2625 . . 3 (𝜓 → (((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅))))
371, 2, 3, 4, 5, 6, 7bnj1256 30337 . . . . . . 7 (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)
384bnj1292 30140 . . . . . . . . 9 𝐷 ⊆ dom 𝑔
39 fndm 5904 . . . . . . . . 9 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
4038, 39syl5sseq 3616 . . . . . . . 8 (𝑔 Fn 𝑑𝐷𝑑)
41 fnssres 5918 . . . . . . . 8 ((𝑔 Fn 𝑑𝐷𝑑) → (𝑔𝐷) Fn 𝐷)
4240, 41mpdan 699 . . . . . . 7 (𝑔 Fn 𝑑 → (𝑔𝐷) Fn 𝐷)
4337, 42bnj31 30039 . . . . . 6 (𝜑 → ∃𝑑𝐵 (𝑔𝐷) Fn 𝐷)
4443bnj1265 30137 . . . . 5 (𝜑 → (𝑔𝐷) Fn 𝐷)
457, 44bnj835 30083 . . . 4 (𝜓 → (𝑔𝐷) Fn 𝐷)
461, 2, 3, 4, 5, 6, 7bnj1259 30338 . . . . . . 7 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
474bnj1293 30141 . . . . . . . . 9 𝐷 ⊆ dom
48 fndm 5904 . . . . . . . . 9 ( Fn 𝑑 → dom = 𝑑)
4947, 48syl5sseq 3616 . . . . . . . 8 ( Fn 𝑑𝐷𝑑)
50 fnssres 5918 . . . . . . . 8 (( Fn 𝑑𝐷𝑑) → (𝐷) Fn 𝐷)
5149, 50mpdan 699 . . . . . . 7 ( Fn 𝑑 → (𝐷) Fn 𝐷)
5246, 51bnj31 30039 . . . . . 6 (𝜑 → ∃𝑑𝐵 (𝐷) Fn 𝐷)
5352bnj1265 30137 . . . . 5 (𝜑 → (𝐷) Fn 𝐷)
547, 53bnj835 30083 . . . 4 (𝜓 → (𝐷) Fn 𝐷)
55 fvreseq 6227 . . . 4 ((((𝑔𝐷) Fn 𝐷 ∧ (𝐷) Fn 𝐷) ∧ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷) → (((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧)))
5645, 54, 8, 55syl21anc 1317 . . 3 (𝜓 → (((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧)))
5736, 56bitr3d 269 . 2 (𝜓 → ((𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧)))
5833, 57mpbird 246 1 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = 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 4583  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-bnj17 30006 This theorem is referenced by:  bnj1311  30346
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