Theorem bnj1125 35023

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 35022	. . 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 34959	. . 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 34999	. . 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 34728	. . . . . . . 8 ⊢ ( TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
12		rsp 3230	. . . . . . . 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 34946	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → pred(𝑦, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
17	16	sseq1d 3990	. . . . . . 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 1930	. . . 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 35019	. 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 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   ⊆ wss 3926   predc-bnj14 34719   FrSe w-bnj15 34723   trClc-bnj18 34725   TrFow-bnj19 34727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-reg 9606  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-1o 8480  df-bnj17 34718  df-bnj14 34720  df-bnj13 34722  df-bnj15 34724  df-bnj18 34726  df-bnj19 34728

This theorem is referenced by:  bnj1137  35026  bnj1136  35028  bnj1175  35035  bnj1408  35067  bnj1417  35072  bnj1452  35083