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Theorem bnj1326 30823
Description: Technical lemma for bnj60 30859. 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 2686 . . . 4 (𝑞 = → (𝑞𝐶𝐶))
213anbi3d 1402 . . 3 (𝑞 = → ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶)))
3 dmeq 5286 . . . . . . 7 (𝑞 = → dom 𝑞 = dom )
43ineq2d 3794 . . . . . 6 (𝑞 = → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ))
54reseq2d 5358 . . . . 5 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom )))
6 bnj1326.4 . . . . . 6 𝐷 = (dom 𝑔 ∩ dom )
76reseq2i 5355 . . . . 5 (𝑔𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ))
85, 7syl6eqr 2673 . . . 4 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔𝐷))
94reseq2d 5358 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom )))
10 reseq1 5352 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
119, 10eqtrd 2655 . . . . 5 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = ( ↾ (dom 𝑔 ∩ dom )))
126reseq2i 5355 . . . . 5 (𝐷) = ( ↾ (dom 𝑔 ∩ dom ))
1311, 12syl6eqr 2673 . . . 4 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝐷))
148, 13eqeq12d 2636 . . 3 (𝑞 = → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔𝐷) = (𝐷)))
152, 14imbi12d 334 . 2 (𝑞 = → (((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))))
16 eleq1 2686 . . . . 5 (𝑝 = 𝑔 → (𝑝𝐶𝑔𝐶))
17163anbi2d 1401 . . . 4 (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶)))
18 dmeq 5286 . . . . . . . 8 (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
1918ineq1d 3793 . . . . . . 7 (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
2019reseq2d 5358 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21 reseq1 5352 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2220, 21eqtrd 2655 . . . . 5 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2319reseq2d 5358 . . . . 5 (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
2422, 23eqeq12d 2636 . . . 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 334 . . 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 2621 . . . 4 (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
3026, 27, 28, 29bnj1311 30821 . . 3 ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
3125, 30chvarv 2262 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
3215, 31chvarv 2262 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  cin 3555  wss 3556  cop 4156  dom cdm 5076  cres 5078   Fn wfn 5844  cfv 5849   predc-bnj14 30482   FrSe w-bnj15 30486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-reg 8444  ax-inf2 8485
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-1o 7508  df-bnj17 30481  df-bnj14 30483  df-bnj13 30485  df-bnj15 30487  df-bnj18 30489  df-bnj19 30491
This theorem is referenced by:  bnj1321  30824  bnj1384  30829
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