Theorem bnj1137 33994

Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1137.1	⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj1137	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo(𝐵, 𝐴, 𝑅))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem bnj1137

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1137.1	. . . . . 6 ⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
2	1	eleq2i 2825	. . . . 5 ⊢ (𝑣 ∈ 𝐵 ↔ 𝑣 ∈ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
3		elun 4147	. . . . 5 ⊢ (𝑣 ∈ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ↔ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
4	2, 3	bitri 274	. . . 4 ⊢ (𝑣 ∈ 𝐵 ↔ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
5		bnj213 33881	. . . . . . . . 9 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
6	5	sseli 3977	. . . . . . . 8 ⊢ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) → 𝑣 ∈ 𝐴)
7		bnj906 33929	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑣 ∈ 𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
8	7	adantlr 713	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
9	6, 8	sylan2 593	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
10		bnj906 33929	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
11	10	sselda 3981	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → 𝑣 ∈ trCl(𝑋, 𝐴, 𝑅))
12		bnj18eq1 33926	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → trCl(𝑦, 𝐴, 𝑅) = trCl(𝑣, 𝐴, 𝑅))
13	12	ssiun2s 5050	. . . . . . . 8 ⊢ (𝑣 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑣, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
14	11, 13	syl 17	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → trCl(𝑣, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
15	9, 14	sstrd 3991	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
16		bnj1147 33993	. . . . . . . . . . 11 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
17	16	rgenw 3065	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
18		iunss 5047	. . . . . . . . . 10 ⊢ (∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴 ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
19	17, 18	mpbir 230	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
20	19	sseli 3977	. . . . . . . 8 ⊢ (𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) → 𝑣 ∈ 𝐴)
21	20, 8	sylan2 593	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
22		bnj1125 33991	. . . . . . . . . . . 12 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23	22	3expia 1121	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
24	23	ralrimiv 3145	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25		iunss 5047	. . . . . . . . . 10 ⊢ (∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
26	24, 25	sylibr 233	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
27	26	sselda 3981	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → 𝑣 ∈ trCl(𝑋, 𝐴, 𝑅))
28	27, 13	syl 17	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑣, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
29	21, 28	sstrd 3991	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
30	15, 29	jaodan 956	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) → pred(𝑣, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
31		ssun2 4172	. . . . . 6 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
32	31, 1	sseqtrri 4018	. . . . 5 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐵
33	30, 32	sstrdi 3993	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
34	4, 33	sylan2b 594	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
35	34	ralrimiva 3146	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
36		df-bnj19 33696	. 2 ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑣 ∈ 𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
37	35, 36	sylibr 233	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo(𝐵, 𝐴, 𝑅))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∪ cun 3945   ⊆ wss 3947  ∪ ciun 4996   predc-bnj14 33687   FrSe w-bnj15 33691   trClc-bnj18 33693   TrFow-bnj19 33695

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-reg 9583  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-bnj17 33686  df-bnj14 33688  df-bnj13 33690  df-bnj15 33692  df-bnj18 33694  df-bnj19 33696

This theorem is referenced by:  bnj1136  33996