Theorem bnj1326 30348

Description: Technical lemma for bnj60 30384. 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		eleq1 2676	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ∈ 𝐶 ↔ ℎ ∈ 𝐶))
2	1	3anbi3d 1397	. . 3 ⊢ (𝑞 = ℎ → ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶)))
3		dmeq 5246	. . . . . . 7 ⊢ (𝑞 = ℎ → dom 𝑞 = dom ℎ)
4	3	ineq2d 3776	. . . . . 6 ⊢ (𝑞 = ℎ → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ℎ))
5	4	reseq2d 5317	. . . . 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 5314	. . . . 5 ⊢ (𝑔 ↾ 𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ℎ))
8	5, 7	syl6eqr 2662	. . . 4 ⊢ (𝑞 = ℎ → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ 𝐷))
9	4	reseq2d 5317	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)))
10		reseq1 5311	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
11	9, 10	eqtrd 2644	. . . . 5 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
12	6	reseq2i 5314	. . . . 5 ⊢ (ℎ ↾ 𝐷) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))
13	11, 12	syl6eqr 2662	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ 𝐷))
14	8, 13	eqeq12d 2625	. . 3 ⊢ (𝑞 = ℎ → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷)))
15	2, 14	imbi12d 333	. 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		eleq1 2676	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
17	16	3anbi2d 1396	. . . 4 ⊢ (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶)))
18		dmeq 5246	. . . . . . . 8 ⊢ (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
19	18	ineq1d 3775	. . . . . . 7 ⊢ (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
20	19	reseq2d 5317	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21		reseq1 5311	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
22	20, 21	eqtrd 2644	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
23	19	reseq2d 5317	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
24	22, 23	eqeq12d 2625	. . . 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 333	. . 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 2610	. . . 4 ⊢ (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
30	26, 27, 28, 29	bnj1311 30346	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
31	25, 30	chvarv 2251	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
32	15, 31	chvarv 2251	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804   predc-bnj14 30007   FrSe w-bnj15 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016

This theorem is referenced by:  bnj1321  30349  bnj1384  30354