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Theorem bnj1326 34335
Description: Technical lemma for bnj60 34371. 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 eleq1w 2814 . . . 4 (𝑞 = → (𝑞𝐶𝐶))
213anbi3d 1440 . . 3 (𝑞 = → ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶)))
3 dmeq 5902 . . . . . . 7 (𝑞 = → dom 𝑞 = dom )
43ineq2d 4211 . . . . . 6 (𝑞 = → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ))
54reseq2d 5980 . . . . 5 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom )))
6 bnj1326.4 . . . . . 6 𝐷 = (dom 𝑔 ∩ dom )
76reseq2i 5977 . . . . 5 (𝑔𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ))
85, 7eqtr4di 2788 . . . 4 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔𝐷))
94reseq2d 5980 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom )))
10 reseq1 5974 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
119, 10eqtrd 2770 . . . . 5 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = ( ↾ (dom 𝑔 ∩ dom )))
126reseq2i 5977 . . . . 5 (𝐷) = ( ↾ (dom 𝑔 ∩ dom ))
1311, 12eqtr4di 2788 . . . 4 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝐷))
148, 13eqeq12d 2746 . . 3 (𝑞 = → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔𝐷) = (𝐷)))
152, 14imbi12d 343 . 2 (𝑞 = → (((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))))
16 eleq1w 2814 . . . . 5 (𝑝 = 𝑔 → (𝑝𝐶𝑔𝐶))
17163anbi2d 1439 . . . 4 (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶)))
18 dmeq 5902 . . . . . . . 8 (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
1918ineq1d 4210 . . . . . . 7 (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
2019reseq2d 5980 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21 reseq1 5974 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2220, 21eqtrd 2770 . . . . 5 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2319reseq2d 5980 . . . . 5 (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
2422, 23eqeq12d 2746 . . . 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 343 . . 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 2730 . . . 4 (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
3026, 27, 28, 29bnj1311 34333 . . 3 ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
3125, 30chvarvv 2000 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
3215, 31chvarvv 2000 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  {cab 2707  wral 3059  wrex 3068  cin 3946  wss 3947  cop 4633  dom cdm 5675  cres 5677   Fn wfn 6537  cfv 6542   predc-bnj14 33997   FrSe w-bnj15 34001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-bnj17 33996  df-bnj14 33998  df-bnj13 34000  df-bnj15 34002  df-bnj18 34004  df-bnj19 34006
This theorem is referenced by:  bnj1321  34336  bnj1384  34341
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