Theorem bnj1326 35639

Description: Technical lemma for bnj60 35675. 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 2844	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ∈ 𝐶 ↔ ℎ ∈ 𝐶))
2	1	3anbi3d 1470	. . 3 ⊢ (𝑞 = ℎ → ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶)))
3		dmeq 5885	. . . . . . 7 ⊢ (𝑞 = ℎ → dom 𝑞 = dom ℎ)
4	3	ineq2d 4166	. . . . . 6 ⊢ (𝑞 = ℎ → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ℎ))
5	4	reseq2d 5970	. . . . 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 5967	. . . . 5 ⊢ (𝑔 ↾ 𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ℎ))
8	5, 7	eqtr4di 2814	. . . 4 ⊢ (𝑞 = ℎ → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ 𝐷))
9	4	reseq2d 5970	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)))
10		reseq1 5964	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
11	9, 10	eqtrd 2796	. . . . 5 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
12	6	reseq2i 5967	. . . . 5 ⊢ (ℎ ↾ 𝐷) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))
13	11, 12	eqtr4di 2814	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ 𝐷))
14	8, 13	eqeq12d 2777	. . 3 ⊢ (𝑞 = ℎ → ((𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) ↔ (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷)))
15	2, 14	imbi12d 347	. 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 2844	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
17	16	3anbi2d 1469	. . . 4 ⊢ (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶)))
18		dmeq 5885	. . . . . . . 8 ⊢ (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
19	18	ineq1d 4165	. . . . . . 7 ⊢ (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
20	19	reseq2d 5970	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21		reseq1 5964	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
22	20, 21	eqtrd 2796	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
23	19	reseq2d 5970	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
24	22, 23	eqeq12d 2777	. . . 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 347	. . 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 2761	. . . 4 ⊢ (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
30	26, 27, 28, 29	bnj1311 35637	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
31	25, 30	chvarvv 2022	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
32	15, 31	chvarvv 2022	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4590  dom cdm 5651   ↾ cres 5653   Fn wfn 6526  ‘cfv 6531   predc-bnj14 35302   FrSe w-bnj15 35306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-reg 9570  ax-inf2 9626

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7867  df-1o 8460  df-bnj17 35301  df-bnj14 35303  df-bnj13 35305  df-bnj15 35307  df-bnj18 35309  df-bnj19 35311

This theorem is used by:  bnj1321  35640  bnj1384  35645