Theorem bnj1326 32906

Description: Technical lemma for bnj60 32942. 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 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1326.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1326.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1326.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)

Assertion

Ref	Expression
bnj1326	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓   𝑅,𝑑,𝑓,𝑥

Allowed substitution hints:   𝐴(𝑔,ℎ)   𝐵(𝑥,𝑔,ℎ,𝑑)   𝐶(𝑥,𝑓,𝑔,ℎ,𝑑)   𝐷(𝑥,𝑓,𝑔,ℎ,𝑑)   𝑅(𝑔,ℎ)   𝐺(𝑥,𝑔,ℎ)   𝑌(𝑥,𝑓,𝑔,ℎ,𝑑)

Proof of Theorem bnj1326

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2821	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ∈ 𝐶 ↔ ℎ ∈ 𝐶))
2	1	3anbi3d 1440	. . 3 ⊢ (𝑞 = ℎ → ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶)))
3		dmeq 5801	. . . . . . 7 ⊢ (𝑞 = ℎ → dom 𝑞 = dom ℎ)
4	3	ineq2d 4143	. . . . . 6 ⊢ (𝑞 = ℎ → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ℎ))
5	4	reseq2d 5880	. . . . 5 ⊢ (𝑞 = ℎ → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)))
6		bnj1326.4	. . . . . 6 ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
7	6	reseq2i 5877	. . . . 5 ⊢ (𝑔 ↾ 𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ℎ))
8	5, 7	eqtr4di 2797	. . . 4 ⊢ (𝑞 = ℎ → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ 𝐷))
9	4	reseq2d 5880	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)))
10		reseq1 5874	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
11	9, 10	eqtrd 2778	. . . . 5 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
12	6	reseq2i 5877	. . . . 5 ⊢ (ℎ ↾ 𝐷) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))
13	11, 12	eqtr4di 2797	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ 𝐷))
14	8, 13	eqeq12d 2754	. . 3 ⊢ (𝑞 = ℎ → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷)))
15	2, 14	imbi12d 344	. 2 ⊢ (𝑞 = ℎ → (((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))))
16		eleq1w 2821	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
17	16	3anbi2d 1439	. . . 4 ⊢ (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶)))
18		dmeq 5801	. . . . . . . 8 ⊢ (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
19	18	ineq1d 4142	. . . . . . 7 ⊢ (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
20	19	reseq2d 5880	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21		reseq1 5874	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
22	20, 21	eqtrd 2778	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
23	19	reseq2d 5880	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
24	22, 23	eqeq12d 2754	. . . 4 ⊢ (𝑝 = 𝑔 → ((𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) ↔ (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞))))
25	17, 24	imbi12d 344	. . 3 ⊢ (𝑝 = 𝑔 → (((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))))
26		bnj1326.1	. . . 4 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
27		bnj1326.2	. . . 4 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
28		bnj1326.3	. . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
29		eqid 2738	. . . 4 ⊢ (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
30	26, 27, 28, 29	bnj1311 32904	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
31	25, 30	chvarvv 2003	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
32	15, 31	chvarvv 2003	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  ⟨cop 4564  dom cdm 5580   ↾ cres 5582   Fn wfn 6413  ‘cfv 6418   predc-bnj14 32567   FrSe w-bnj15 32571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-bnj17 32566  df-bnj14 32568  df-bnj13 32570  df-bnj15 32572  df-bnj18 32574  df-bnj19 32576

This theorem is referenced by:  bnj1321  32907  bnj1384  32912