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Theorem bnj1253 31423
Description: Technical lemma for bnj60 31468. 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 31218 . . 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 31421 . . . . . . . . . 10 (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)
106bnj1292 31224 . . . . . . . . . . . 12 𝐷 ⊆ dom 𝑔
11 fndm 6130 . . . . . . . . . . . 12 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1210, 11syl5sseq 3802 . . . . . . . . . . 11 (𝑔 Fn 𝑑𝐷𝑑)
13 fnssres 6144 . . . . . . . . . . 11 ((𝑔 Fn 𝑑𝐷𝑑) → (𝑔𝐷) Fn 𝐷)
1412, 13mpdan 667 . . . . . . . . . 10 (𝑔 Fn 𝑑 → (𝑔𝐷) Fn 𝐷)
159, 14bnj31 31125 . . . . . . . . 9 (𝜑 → ∃𝑑𝐵 (𝑔𝐷) Fn 𝐷)
1615bnj1265 31221 . . . . . . . 8 (𝜑 → (𝑔𝐷) Fn 𝐷)
173, 4, 5, 6, 7, 1, 8bnj1259 31422 . . . . . . . . . 10 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
186bnj1293 31225 . . . . . . . . . . . 12 𝐷 ⊆ dom
19 fndm 6130 . . . . . . . . . . . 12 ( Fn 𝑑 → dom = 𝑑)
2018, 19syl5sseq 3802 . . . . . . . . . . 11 ( Fn 𝑑𝐷𝑑)
21 fnssres 6144 . . . . . . . . . . 11 (( Fn 𝑑𝐷𝑑) → (𝐷) Fn 𝐷)
2220, 21mpdan 667 . . . . . . . . . 10 ( Fn 𝑑 → (𝐷) Fn 𝐷)
2317, 22bnj31 31125 . . . . . . . . 9 (𝜑 → ∃𝑑𝐵 (𝐷) Fn 𝐷)
2423bnj1265 31221 . . . . . . . 8 (𝜑 → (𝐷) Fn 𝐷)
25 ssid 3773 . . . . . . . . 9 𝐷𝐷
26 fvreseq 6462 . . . . . . . . 9 ((((𝑔𝐷) Fn 𝐷 ∧ (𝐷) Fn 𝐷) ∧ 𝐷𝐷) → (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ ∀𝑥𝐷 ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)))
2725, 26mpan2 671 . . . . . . . 8 (((𝑔𝐷) Fn 𝐷 ∧ (𝐷) Fn 𝐷) → (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ ∀𝑥𝐷 ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)))
2816, 24, 27syl2anc 573 . . . . . . 7 (𝜑 → (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ ∀𝑥𝐷 ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)))
29 residm 5571 . . . . . . . 8 ((𝑔𝐷) ↾ 𝐷) = (𝑔𝐷)
30 residm 5571 . . . . . . . 8 ((𝐷) ↾ 𝐷) = (𝐷)
3129, 30eqeq12i 2785 . . . . . . 7 (((𝑔𝐷) ↾ 𝐷) = ((𝐷) ↾ 𝐷) ↔ (𝑔𝐷) = (𝐷))
32 df-ral 3066 . . . . . . 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 6348 . . . . . . . . 9 (𝑥𝐷 → ((𝑔𝐷)‘𝑥) = (𝑔𝑥))
35 fvres 6348 . . . . . . . . 9 (𝑥𝐷 → ((𝐷)‘𝑥) = (𝑥))
3634, 35eqeq12d 2786 . . . . . . . 8 (𝑥𝐷 → (((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥) ↔ (𝑔𝑥) = (𝑥)))
3736pm5.74i 260 . . . . . . 7 ((𝑥𝐷 → ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)) ↔ (𝑥𝐷 → (𝑔𝑥) = (𝑥)))
3837albii 1895 . . . . . 6 (∀𝑥(𝑥𝐷 → ((𝑔𝐷)‘𝑥) = ((𝐷)‘𝑥)) ↔ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥)))
3933, 38syl6bb 276 . . . . 5 (𝜑 → ((𝑔𝐷) = (𝐷) ↔ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥))))
4039necon3abid 2979 . . . 4 (𝜑 → ((𝑔𝐷) ≠ (𝐷) ↔ ¬ ∀𝑥(𝑥𝐷 → (𝑔𝑥) = (𝑥))))
41 df-rex 3067 . . . . 5 (∃𝑥𝐷 (𝑔𝑥) ≠ (𝑥) ↔ ∃𝑥(𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)))
42 pm4.61 391 . . . . . . 7 (¬ (𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ (𝑥𝐷 ∧ ¬ (𝑔𝑥) = (𝑥)))
43 df-ne 2944 . . . . . . . 8 ((𝑔𝑥) ≠ (𝑥) ↔ ¬ (𝑔𝑥) = (𝑥))
4443anbi2i 609 . . . . . . 7 ((𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)) ↔ (𝑥𝐷 ∧ ¬ (𝑔𝑥) = (𝑥)))
4542, 44bitr4i 267 . . . . . 6 (¬ (𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ (𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)))
4645exbii 1924 . . . . 5 (∃𝑥 ¬ (𝑥𝐷 → (𝑔𝑥) = (𝑥)) ↔ ∃𝑥(𝑥𝐷 ∧ (𝑔𝑥) ≠ (𝑥)))
47 exnal 1902 . . . . 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 3007 . . 3 (𝐸 ≠ ∅ ↔ {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)} ≠ ∅)
52 rabn0 4104 . . 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 382  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  {crab 3065  cin 3722  wss 3723  c0 4063  cop 4322   class class class wbr 4786  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031  w-bnj17 31092   predc-bnj14 31094   FrSe w-bnj15 31098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-bnj17 31093
This theorem is referenced by:  bnj1311  31430
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