Theorem bnj1125 34989

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 34988	. . 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 34925	. . 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 34965	. . 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 2811	. . . . 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 34694	. . . . . . . 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 2817	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)))
16		bnj602 34912	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → pred(𝑦, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
17	16	sseq1d 3981	. . . . . . 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 34985	. 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 2109  ∀wral 3045  Vcvv 3450   ⊆ wss 3917   predc-bnj14 34685   FrSe w-bnj15 34689   trClc-bnj18 34691   TrFow-bnj19 34693

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-om 7846  df-1o 8437  df-bnj17 34684  df-bnj14 34686  df-bnj13 34688  df-bnj15 34690  df-bnj18 34692  df-bnj19 34694

This theorem is referenced by:  bnj1137  34992  bnj1136  34994  bnj1175  35001  bnj1408  35033  bnj1417  35038  bnj1452  35049