Theorem bnj1125 35154

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 35153	. . 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 35090	. . 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 35130	. . 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 2819	. . . . 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 34860	. . . . . . . 8 ⊢ ( TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
12		rsp 3226	. . . . . . . 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 2825	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)))
16		bnj602 35077	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → pred(𝑦, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
17	16	sseq1d 3954	. . . . . . 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 35150	. 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 3052  Vcvv 3430   ⊆ wss 3890   predc-bnj14 34851   FrSe w-bnj15 34855   trClc-bnj18 34857   TrFow-bnj19 34859

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-reg 9502  ax-inf2 9557

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-om 7813  df-1o 8400  df-bnj17 34850  df-bnj14 34852  df-bnj13 34854  df-bnj15 34856  df-bnj18 34858  df-bnj19 34860

This theorem is referenced by:  bnj1137  35157  bnj1136  35159  bnj1175  35166  bnj1408  35198  bnj1417  35203  bnj1452  35214