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Theorem bnj1253 31211
Description: Technical lemma for bnj60 31256. 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
bnj1253.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1253.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1253.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1253.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1253.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1253.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1253.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
Assertion
Ref Expression
bnj1253 (𝜑𝐸 ≠ ∅)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓,𝑔   𝐵,,𝑓   𝐷,𝑑   𝑥,𝐷   𝑓,𝐺,𝑔   ,𝐺   𝑅,𝑓   𝑔,𝑌   ,𝑌   𝑓,𝑑,𝑔   ,𝑑   𝑥,𝑓,𝑔   𝑥,
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑔,,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑦,𝑓,𝑔,)   𝑅(𝑥,𝑦,𝑔,,𝑑)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1253
StepHypRef Expression
1 bnj1253.6 . . . 4 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
21bnj1254 31006 . . 3 (𝜑 → (𝑔𝐷) ≠ (𝐷))
3 bnj1253.1 . . . . . . . . . . 11 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4 bnj1253.2 . . . . . . . . . . 11 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
5 bnj1253.3 . . . . . . . . . . 11 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
6 bnj1253.4 . . . . . . . . . . 11 𝐷 = (dom 𝑔 ∩ dom )
7 bnj1253.5 . . . . . . . . . . 11 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
8 bnj1253.7 . . . . . . . . . . 11 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
93, 4, 5, 6, 7, 1, 8bnj1256 31209 . . . . . . . . . 10 (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)
106bnj1292 31012 . . . . . . . . . . . 12 𝐷 ⊆ dom 𝑔
11 fndm 6028 . . . . . . . . . . . 12 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1210, 11syl5sseq 3686 . . . . . . . . . . 11 (𝑔 Fn 𝑑𝐷𝑑)
13 fnssres 6042 . . . . . . . . . . 11 ((𝑔 Fn 𝑑𝐷𝑑) → (𝑔𝐷) Fn 𝐷)
1412, 13mpdan 703 . . . . . . . . . 10 (𝑔 Fn 𝑑 → (𝑔𝐷) Fn 𝐷)
159, 14bnj31 30913 . . . . . . . . 9 (𝜑 → ∃𝑑𝐵 (𝑔𝐷) Fn 𝐷)
1615bnj1265 31009 . . . . . . . 8 (𝜑 → (𝑔𝐷) Fn 𝐷)
173, 4, 5, 6, 7, 1, 8bnj1259 31210 . . . . . . . . . 10 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
186bnj1293 31013 . . . . . . . . . . . 12 𝐷 ⊆ dom
19 fndm 6028 . . . . . . . . . . . 12 ( Fn 𝑑 → dom = 𝑑)
2018, 19syl5sseq 3686 . . . . . . . . . . 11 ( Fn 𝑑𝐷𝑑)
21 fnssres 6042 . . . . . . . . . . 11 (( Fn 𝑑𝐷𝑑) → (𝐷) Fn 𝐷)
2220, 21mpdan 703 . . . . . . . . . 10 ( Fn 𝑑 → (𝐷) Fn 𝐷)
2317, 22bnj31 30913 . . . . . . . . 9 (𝜑 → ∃𝑑𝐵 (𝐷) Fn 𝐷)
2423bnj1265 31009 . . . . . . . 8 (𝜑 → (𝐷) Fn 𝐷)
25 ssid 3657 . . . . . . . . 9 𝐷𝐷
26 fvreseq 6359 . . . . . . . . 9 ((((𝑔𝐷) Fn 𝐷 ∧ (𝐷) Fn 𝐷) ∧ 𝐷𝐷) → (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ ∀𝑥𝐷 ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)))
2725, 26mpan2 707 . . . . . . . 8 (((𝑔𝐷) Fn 𝐷 ∧ (𝐷) Fn 𝐷) → (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ ∀𝑥𝐷 ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)))
2816, 24, 27syl2anc 694 . . . . . . 7 (𝜑 → (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ ∀𝑥𝐷 ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)))
29 residm 5465 . . . . . . . 8 ((𝑔𝐷) ↾ 𝐷) = (𝑔𝐷)
30 residm 5465 . . . . . . . 8 ((𝐷) ↾ 𝐷) = (𝐷)
3129, 30eqeq12i 2665 . . . . . . 7 (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ (𝑔𝐷) = (𝐷))
32 df-ral 2946 . . . . . . 7 (∀𝑥𝐷 ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥) ↔ ∀𝑥(𝑥𝐷 → ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)))
3328, 31, 323bitr3g 302 . . . . . 6 (𝜑 → ((𝑔𝐷) = (𝐷) ↔ ∀𝑥(𝑥𝐷 → ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥))))
34 fvres 6245 . . . . . . . . 9 (𝑥𝐷 → ((𝑔𝐷)‘𝑥) = (𝑔𝑥))
35 fvres 6245 . . . . . . . . 9 (𝑥𝐷 → ((𝐷)‘𝑥) = (𝑥))
3634, 35eqeq12d 2666 . . . . . . . 8 (𝑥𝐷 → (((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥) ↔ (𝑔𝑥) = (𝑥)))
3736pm5.74i 260 . . . . . . 7 ((𝑥𝐷 → ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)) ↔ (𝑥𝐷 → (𝑔𝑥) = (𝑥)))
3837albii 1787 . . . . . 6 (∀𝑥(𝑥𝐷 → ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)) ↔ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥)))
3933, 38syl6bb 276 . . . . 5 (𝜑 → ((𝑔𝐷) = (𝐷) ↔ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥))))
4039necon3abid 2859 . . . 4 (𝜑 → ((𝑔𝐷) ≠ (𝐷) ↔ ¬ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥))))
41 df-rex 2947 . . . . 5 (∃𝑥𝐷 (𝑔𝑥) ≠ (𝑥) ↔ ∃𝑥(𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)))
42 pm4.61 441 . . . . . . 7 (¬ (𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ (𝑥𝐷 ∧ ¬ (𝑔𝑥) = (𝑥)))
43 df-ne 2824 . . . . . . . 8 ((𝑔𝑥) ≠ (𝑥) ↔ ¬ (𝑔𝑥) = (𝑥))
4443anbi2i 730 . . . . . . 7 ((𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)) ↔ (𝑥𝐷 ∧ ¬ (𝑔𝑥) = (𝑥)))
4542, 44bitr4i 267 . . . . . 6 (¬ (𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ (𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)))
4645exbii 1814 . . . . 5 (∃𝑥 ¬ (𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ ∃𝑥(𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)))
47 exnal 1794 . . . . 5 (∃𝑥 ¬ (𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ ¬ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥)))
4841, 46, 473bitr2ri 289 . . . 4 (¬ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ ∃𝑥𝐷 (𝑔𝑥) ≠ (𝑥))
4940, 48syl6bb 276 . . 3 (𝜑 → ((𝑔𝐷) ≠ (𝐷) ↔ ∃𝑥𝐷 (𝑔𝑥) ≠ (𝑥)))
502, 49mpbid 222 . 2 (𝜑 → ∃𝑥𝐷 (𝑔𝑥) ≠ (𝑥))
517neeq1i 2887 . . 3 (𝐸 ≠ ∅ ↔ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅)
52 rabn0 3991 . . 3 ({𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅ ↔ ∃𝑥𝐷 (𝑔𝑥) ≠ (𝑥))
5351, 52bitri 264 . 2 (𝐸 ≠ ∅ ↔ ∃𝑥𝐷 (𝑔𝑥) ≠ (𝑥))
5450, 53sylibr 224 1 (𝜑𝐸 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  cin 3606  wss 3607  c0 3948  cop 4216   class class class wbr 4685  dom cdm 5143  cres 5145   Fn wfn 5921  cfv 5926  w-bnj17 30880   predc-bnj14 30882   FrSe w-bnj15 30886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-bnj17 30881
This theorem is referenced by:  bnj1311  31218
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