Theorem bnj1137 35177

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 2829	. . . . 5 ⊢ (𝑣 ∈ 𝐵 ↔ 𝑣 ∈ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
3		elun 4107	. . . . 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 275	. . . 4 ⊢ (𝑣 ∈ 𝐵 ↔ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
5		bnj213 35064	. . . . . . . . 9 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
6	5	sseli 3931	. . . . . . . 8 ⊢ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) → 𝑣 ∈ 𝐴)
7		bnj906 35112	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑣 ∈ 𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
8	7	adantlr 716	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
9	6, 8	sylan2 594	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
10		bnj906 35112	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
11	10	sselda 3935	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → 𝑣 ∈ trCl(𝑋, 𝐴, 𝑅))
12		bnj18eq1 35109	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → trCl(𝑦, 𝐴, 𝑅) = trCl(𝑣, 𝐴, 𝑅))
13	12	ssiun2s 5006	. . . . . . . 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 3946	. . . . . 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 35176	. . . . . . . . . . 11 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
17	16	rgenw 3056	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
18		iunss 5002	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
20	19	sseli 3931	. . . . . . . 8 ⊢ (𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) → 𝑣 ∈ 𝐴)
21	20, 8	sylan2 594	. . . . . . 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 35174	. . . . . . . . . . . 12 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23	22	3expia 1122	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
24	23	ralrimiv 3129	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25		iunss 5002	. . . . . . . . . 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 234	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
27	26	sselda 3935	. . . . . . . 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 3946	. . . . . 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 960	. . . . 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 4133	. . . . . 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 3985	. . . . 5 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐵
33	30, 32	sstrdi 3948	. . . 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 595	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
35	34	ralrimiva 3130	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
36		df-bnj19 34880	. 2 ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑣 ∈ 𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
37	35, 36	sylibr 234	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo(𝐵, 𝐴, 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3901   ⊆ wss 3903  ∪ ciun 4948   predc-bnj14 34871   FrSe w-bnj15 34875   trClc-bnj18 34877   TrFow-bnj19 34879

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-reg 9511  ax-inf2 9564

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-om 7821  df-1o 8409  df-bnj17 34870  df-bnj14 34872  df-bnj13 34874  df-bnj15 34876  df-bnj18 34878  df-bnj19 34880

This theorem is referenced by:  bnj1136  35179