Theorem bnj1326 35019

Description: Technical lemma for bnj60 35055. 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 2822	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ∈ 𝐶 ↔ ℎ ∈ 𝐶))
2	1	3anbi3d 1441	. . 3 ⊢ (𝑞 = ℎ → ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶)))
3		dmeq 5917	. . . . . . 7 ⊢ (𝑞 = ℎ → dom 𝑞 = dom ℎ)
4	3	ineq2d 4228	. . . . . 6 ⊢ (𝑞 = ℎ → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ℎ))
5	4	reseq2d 6000	. . . . 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 5997	. . . . 5 ⊢ (𝑔 ↾ 𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ℎ))
8	5, 7	eqtr4di 2793	. . . 4 ⊢ (𝑞 = ℎ → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ 𝐷))
9	4	reseq2d 6000	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)))
10		reseq1 5994	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
11	9, 10	eqtrd 2775	. . . . 5 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
12	6	reseq2i 5997	. . . . 5 ⊢ (ℎ ↾ 𝐷) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))
13	11, 12	eqtr4di 2793	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ 𝐷))
14	8, 13	eqeq12d 2751	. . 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 2822	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
17	16	3anbi2d 1440	. . . 4 ⊢ (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶)))
18		dmeq 5917	. . . . . . . 8 ⊢ (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
19	18	ineq1d 4227	. . . . . . 7 ⊢ (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
20	19	reseq2d 6000	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21		reseq1 5994	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
22	20, 21	eqtrd 2775	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
23	19	reseq2d 6000	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
24	22, 23	eqeq12d 2751	. . . 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 2735	. . . 4 ⊢ (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
30	26, 27, 28, 29	bnj1311 35017	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
31	25, 30	chvarvv 1996	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
32	15, 31	chvarvv 1996	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068   ∩ cin 3962   ⊆ wss 3963  ⟨cop 4637  dom cdm 5689   ↾ cres 5691   Fn wfn 6558  ‘cfv 6563   predc-bnj14 34681   FrSe w-bnj15 34685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-bnj17 34680  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686  df-bnj18 34688  df-bnj19 34690

This theorem is referenced by:  bnj1321  35020  bnj1384  35025