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Theorem bnj1326 33695
Description: Technical lemma for bnj60 33731. 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 2817 . . . 4 (𝑞 = → (𝑞𝐶𝐶))
213anbi3d 1443 . . 3 (𝑞 = → ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶)))
3 dmeq 5860 . . . . . . 7 (𝑞 = → dom 𝑞 = dom )
43ineq2d 4173 . . . . . 6 (𝑞 = → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ))
54reseq2d 5938 . . . . 5 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom )))
6 bnj1326.4 . . . . . 6 𝐷 = (dom 𝑔 ∩ dom )
76reseq2i 5935 . . . . 5 (𝑔𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ))
85, 7eqtr4di 2791 . . . 4 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔𝐷))
94reseq2d 5938 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom )))
10 reseq1 5932 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
119, 10eqtrd 2773 . . . . 5 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = ( ↾ (dom 𝑔 ∩ dom )))
126reseq2i 5935 . . . . 5 (𝐷) = ( ↾ (dom 𝑔 ∩ dom ))
1311, 12eqtr4di 2791 . . . 4 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝐷))
148, 13eqeq12d 2749 . . 3 (𝑞 = → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔𝐷) = (𝐷)))
152, 14imbi12d 345 . 2 (𝑞 = → (((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))))
16 eleq1w 2817 . . . . 5 (𝑝 = 𝑔 → (𝑝𝐶𝑔𝐶))
17163anbi2d 1442 . . . 4 (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶)))
18 dmeq 5860 . . . . . . . 8 (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
1918ineq1d 4172 . . . . . . 7 (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
2019reseq2d 5938 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21 reseq1 5932 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2220, 21eqtrd 2773 . . . . 5 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2319reseq2d 5938 . . . . 5 (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
2422, 23eqeq12d 2749 . . . 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 345 . . 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 2733 . . . 4 (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
3026, 27, 28, 29bnj1311 33693 . . 3 ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
3125, 30chvarvv 2003 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
3215, 31chvarvv 2003 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  cin 3910  wss 3911  cop 4593  dom cdm 5634  cres 5636   Fn wfn 6492  cfv 6497   predc-bnj14 33357   FrSe w-bnj15 33361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-bnj17 33356  df-bnj14 33358  df-bnj13 33360  df-bnj15 33362  df-bnj18 33364  df-bnj19 33366
This theorem is referenced by:  bnj1321  33696  bnj1384  33701
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