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Theorem bnj1326 31220
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
bnj1326.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1326.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1326.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1326.4 𝐷 = (dom 𝑔 ∩ dom )
Assertion
Ref Expression
bnj1326 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓   𝑅,𝑑,𝑓,𝑥
Allowed substitution hints:   𝐴(𝑔,)   𝐵(𝑥,𝑔,,𝑑)   𝐶(𝑥,𝑓,𝑔,,𝑑)   𝐷(𝑥,𝑓,𝑔,,𝑑)   𝑅(𝑔,)   𝐺(𝑥,𝑔,)   𝑌(𝑥,𝑓,𝑔,,𝑑)

Proof of Theorem bnj1326
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2718 . . . 4 (𝑞 = → (𝑞𝐶𝐶))
213anbi3d 1445 . . 3 (𝑞 = → ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶)))
3 dmeq 5356 . . . . . . 7 (𝑞 = → dom 𝑞 = dom )
43ineq2d 3847 . . . . . 6 (𝑞 = → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ))
54reseq2d 5428 . . . . 5 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom )))
6 bnj1326.4 . . . . . 6 𝐷 = (dom 𝑔 ∩ dom )
76reseq2i 5425 . . . . 5 (𝑔𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ))
85, 7syl6eqr 2703 . . . 4 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔𝐷))
94reseq2d 5428 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom )))
10 reseq1 5422 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
119, 10eqtrd 2685 . . . . 5 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = ( ↾ (dom 𝑔 ∩ dom )))
126reseq2i 5425 . . . . 5 (𝐷) = ( ↾ (dom 𝑔 ∩ dom ))
1311, 12syl6eqr 2703 . . . 4 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝐷))
148, 13eqeq12d 2666 . . 3 (𝑞 = → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔𝐷) = (𝐷)))
152, 14imbi12d 333 . 2 (𝑞 = → (((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))))
16 eleq1 2718 . . . . 5 (𝑝 = 𝑔 → (𝑝𝐶𝑔𝐶))
17163anbi2d 1444 . . . 4 (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶)))
18 dmeq 5356 . . . . . . . 8 (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
1918ineq1d 3846 . . . . . . 7 (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
2019reseq2d 5428 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21 reseq1 5422 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2220, 21eqtrd 2685 . . . . 5 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2319reseq2d 5428 . . . . 5 (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
2422, 23eqeq12d 2666 . . . 4 (𝑝 = 𝑔 → ((𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) ↔ (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))))
2517, 24imbi12d 333 . . 3 (𝑝 = 𝑔 → (((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))))
26 bnj1326.1 . . . 4 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
27 bnj1326.2 . . . 4 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
28 bnj1326.3 . . . 4 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
29 eqid 2651 . . . 4 (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
3026, 27, 28, 29bnj1311 31218 . . 3 ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
3125, 30chvarv 2299 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
3215, 31chvarv 2299 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  cin 3606  wss 3607  cop 4216  dom cdm 5143  cres 5145   Fn wfn 5921  cfv 5926   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-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  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-reu 2948  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-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  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-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-bnj17 30881  df-bnj14 30883  df-bnj13 30885  df-bnj15 30887  df-bnj18 30889  df-bnj19 30891
This theorem is referenced by:  bnj1321  31221  bnj1384  31226
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