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Theorem bnj1326 32406
Description: Technical lemma for bnj60 32442. 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 2875 . . . 4 (𝑞 = → (𝑞𝐶𝐶))
213anbi3d 1439 . . 3 (𝑞 = → ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶)))
3 dmeq 5740 . . . . . . 7 (𝑞 = → dom 𝑞 = dom )
43ineq2d 4142 . . . . . 6 (𝑞 = → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ))
54reseq2d 5822 . . . . 5 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom )))
6 bnj1326.4 . . . . . 6 𝐷 = (dom 𝑔 ∩ dom )
76reseq2i 5819 . . . . 5 (𝑔𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ))
85, 7eqtr4di 2854 . . . 4 (𝑞 = → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔𝐷))
94reseq2d 5822 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom )))
10 reseq1 5816 . . . . . 6 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
119, 10eqtrd 2836 . . . . 5 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = ( ↾ (dom 𝑔 ∩ dom )))
126reseq2i 5819 . . . . 5 (𝐷) = ( ↾ (dom 𝑔 ∩ dom ))
1311, 12eqtr4di 2854 . . . 4 (𝑞 = → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝐷))
148, 13eqeq12d 2817 . . 3 (𝑞 = → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔𝐷) = (𝐷)))
152, 14imbi12d 348 . 2 (𝑞 = → (((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))))
16 eleq1w 2875 . . . . 5 (𝑝 = 𝑔 → (𝑝𝐶𝑔𝐶))
17163anbi2d 1438 . . . 4 (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) ↔ (𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶)))
18 dmeq 5740 . . . . . . . 8 (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
1918ineq1d 4141 . . . . . . 7 (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
2019reseq2d 5822 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21 reseq1 5816 . . . . . 6 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2220, 21eqtrd 2836 . . . . 5 (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
2319reseq2d 5822 . . . . 5 (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
2422, 23eqeq12d 2817 . . . 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 348 . . 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 2801 . . . 4 (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
3026, 27, 28, 29bnj1311 32404 . . 3 ((𝑅 FrSe 𝐴𝑝𝐶𝑞𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
3125, 30chvarvv 2005 . 2 ((𝑅 FrSe 𝐴𝑔𝐶𝑞𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
3215, 31chvarvv 2005 1 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  cin 3883  wss 3884  cop 4534  dom cdm 5523  cres 5525   Fn wfn 6323  cfv 6328   predc-bnj14 32066   FrSe w-bnj15 32070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-bnj17 32065  df-bnj14 32067  df-bnj13 32069  df-bnj15 32071  df-bnj18 32073  df-bnj19 32075
This theorem is referenced by:  bnj1321  32407  bnj1384  32412
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