Theorem bnj1125 35097

Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1125	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Proof of Theorem bnj1125

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
2		bnj1127 35096	. . 3 ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴)
3	2	3ad2ant3 1135	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑌 ∈ 𝐴)
4		bnj893 35033	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
5	4	3adant3 1132	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
6		bnj1029 35073	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
7	6	3adant3 1132	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
8		simp3 1138	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅))
9		elisset 2816	. . . . 5 ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑦 𝑦 = 𝑌)
10	9	3ad2ant3 1135	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → ∃𝑦 𝑦 = 𝑌)
11		df-bnj19 34802	. . . . . . . 8 ⊢ ( TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
12		rsp 3222	. . . . . . . 8 ⊢ (∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
13	11, 12	sylbi 217	. . . . . . 7 ⊢ ( TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
14	7, 13	syl 17	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
15		eleq1 2822	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)))
16		bnj602 35020	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → pred(𝑦, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
17	16	sseq1d 3963	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ( pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
18	15, 17	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))))
19	14, 18	imbitrid 244	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))))
20	19	exlimiv 1931	. . . 4 ⊢ (∃𝑦 𝑦 = 𝑌 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))))
21	10, 20	mpcom 38	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
22	8, 21	mpd 15	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23		biid 261	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑌 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑌 ∈ 𝐴))
24		biid 261	. . 3 ⊢ (( trCl(𝑋, 𝐴, 𝑅) ∈ V ∧ TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ∧ pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ( trCl(𝑋, 𝐴, 𝑅) ∈ V ∧ TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ∧ pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
25	23, 24	bnj1124 35093	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ ( trCl(𝑋, 𝐴, 𝑅) ∈ V ∧ TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ∧ pred(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))) → trCl(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
26	1, 3, 5, 7, 22, 25	syl23anc 1379	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ⊆ wss 3899   predc-bnj14 34793   FrSe w-bnj15 34797   trClc-bnj18 34799   TrFow-bnj19 34801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-om 7807  df-1o 8395  df-bnj17 34792  df-bnj14 34794  df-bnj13 34796  df-bnj15 34798  df-bnj18 34800  df-bnj19 34802

This theorem is referenced by:  bnj1137  35100  bnj1136  35102  bnj1175  35109  bnj1408  35141  bnj1417  35146  bnj1452  35157