Theorem bnj1125 35134

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 1137	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
2		bnj1127 35133	. . 3 ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴)
3	2	3ad2ant3 1136	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑌 ∈ 𝐴)
4		bnj893 35070	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
5	4	3adant3 1133	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
6		bnj1029 35110	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
7	6	3adant3 1133	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
8		simp3 1139	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅))
9		elisset 2818	. . . . 5 ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑦 𝑦 = 𝑌)
10	9	3ad2ant3 1136	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → ∃𝑦 𝑦 = 𝑌)
11		df-bnj19 34840	. . . . . . . 8 ⊢ ( TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
12		rsp 3225	. . . . . . . 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 2824	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)))
16		bnj602 35057	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → pred(𝑦, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
17	16	sseq1d 3953	. . . . . . 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 1932	. . . 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 35130	. 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 1380	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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889   predc-bnj14 34831   FrSe w-bnj15 34835   trClc-bnj18 34837   TrFow-bnj19 34839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-bnj17 34830  df-bnj14 34832  df-bnj13 34834  df-bnj15 34836  df-bnj18 34838  df-bnj19 34840

This theorem is referenced by:  bnj1137  35137  bnj1136  35139  bnj1175  35146  bnj1408  35178  bnj1417  35183  bnj1452  35194