Theorem bnj1326 34981

Description: Technical lemma for bnj60 35017. 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 2816	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ∈ 𝐶 ↔ ℎ ∈ 𝐶))
2	1	3anbi3d 1443	. . 3 ⊢ (𝑞 = ℎ → ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶)))
3		dmeq 5896	. . . . . . 7 ⊢ (𝑞 = ℎ → dom 𝑞 = dom ℎ)
4	3	ineq2d 4202	. . . . . 6 ⊢ (𝑞 = ℎ → (dom 𝑔 ∩ dom 𝑞) = (dom 𝑔 ∩ dom ℎ))
5	4	reseq2d 5979	. . . . 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 5976	. . . . 5 ⊢ (𝑔 ↾ 𝐷) = (𝑔 ↾ (dom 𝑔 ∩ dom ℎ))
8	5, 7	eqtr4di 2787	. . . 4 ⊢ (𝑞 = ℎ → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ 𝐷))
9	4	reseq2d 5979	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)))
10		reseq1 5973	. . . . . 6 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
11	9, 10	eqtrd 2769	. . . . 5 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
12	6	reseq2i 5976	. . . . 5 ⊢ (ℎ ↾ 𝐷) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))
13	11, 12	eqtr4di 2787	. . . 4 ⊢ (𝑞 = ℎ → (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)) = (ℎ ↾ 𝐷))
14	8, 13	eqeq12d 2750	. . 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 2816	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
17	16	3anbi2d 1442	. . . 4 ⊢ (𝑝 = 𝑔 → ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶)))
18		dmeq 5896	. . . . . . . 8 ⊢ (𝑝 = 𝑔 → dom 𝑝 = dom 𝑔)
19	18	ineq1d 4201	. . . . . . 7 ⊢ (𝑝 = 𝑔 → (dom 𝑝 ∩ dom 𝑞) = (dom 𝑔 ∩ dom 𝑞))
20	19	reseq2d 5979	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)))
21		reseq1 5973	. . . . . 6 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
22	20, 21	eqtrd 2769	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)))
23	19	reseq2d 5979	. . . . 5 ⊢ (𝑝 = 𝑔 → (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
24	22, 23	eqeq12d 2750	. . . 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 2734	. . . 4 ⊢ (dom 𝑝 ∩ dom 𝑞) = (dom 𝑝 ∩ dom 𝑞)
30	26, 27, 28, 29	bnj1311 34979	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑝 ↾ (dom 𝑝 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑝 ∩ dom 𝑞)))
31	25, 30	chvarvv 1997	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom 𝑞)) = (𝑞 ↾ (dom 𝑔 ∩ dom 𝑞)))
32	15, 31	chvarvv 1997	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ 𝐷) = (ℎ ↾ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  {cab 2712  ∀wral 3050  ∃wrex 3059   ∩ cin 3932   ⊆ wss 3933  ⟨cop 4614  dom cdm 5667   ↾ cres 5669   Fn wfn 6537  ‘cfv 6542   predc-bnj14 34643   FrSe w-bnj15 34647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-reg 9615  ax-inf2 9664

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7871  df-1o 8489  df-bnj17 34642  df-bnj14 34644  df-bnj13 34646  df-bnj15 34648  df-bnj18 34650  df-bnj19 34652

This theorem is referenced by:  bnj1321  34982  bnj1384  34987