Theorem bnj1137 30317

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 2680	. . . . 5 ⊢ (𝑣 ∈ 𝐵 ↔ 𝑣 ∈ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
3		elun 3715	. . . . 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 263	. . . 4 ⊢ (𝑣 ∈ 𝐵 ↔ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) ∨ 𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
5		bnj213 30206	. . . . . . . . 9 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
6	5	sseli 3564	. . . . . . . 8 ⊢ (𝑣 ∈ pred(𝑋, 𝐴, 𝑅) → 𝑣 ∈ 𝐴)
7		bnj906 30254	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑣 ∈ 𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
8	7	adantlr 747	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ 𝐴) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
9	6, 8	sylan2 490	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → pred(𝑣, 𝐴, 𝑅) ⊆ trCl(𝑣, 𝐴, 𝑅))
10		bnj906 30254	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
11	10	sselda 3568	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ pred(𝑋, 𝐴, 𝑅)) → 𝑣 ∈ trCl(𝑋, 𝐴, 𝑅))
12		bnj18eq1 30251	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → trCl(𝑦, 𝐴, 𝑅) = trCl(𝑣, 𝐴, 𝑅))
13	12	ssiun2s 4500	. . . . . . . 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 3578	. . . . . 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 30316	. . . . . . . . . . 11 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
17	16	rgenw 2908	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
18		iunss 4497	. . . . . . . . . 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 220	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
20	19	sseli 3564	. . . . . . . 8 ⊢ (𝑣 ∈ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) → 𝑣 ∈ 𝐴)
21	20, 8	sylan2 490	. . . . . . 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 30314	. . . . . . . . . . . 12 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23	22	3expia 1259	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
24	23	ralrimiv 2948	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25		iunss 4497	. . . . . . . . . 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 223	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
27	26	sselda 3568	. . . . . . . 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 3578	. . . . . 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 822	. . . . 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 3739	. . . . . 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	sseqtr4i 3601	. . . . 5 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐵
33	30, 32	syl6ss 3580	. . . 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 491	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
35	34	ralrimiva 2949	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
36		df-bnj19 30016	. 2 ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑣 ∈ 𝐵 pred(𝑣, 𝐴, 𝑅) ⊆ 𝐵)
37	35, 36	sylibr 223	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo(𝐵, 𝐴, 𝑅))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538   ⊆ wss 3540  ∪ ciun 4455   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013   TrFow-bnj19 30015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016

This theorem is referenced by:  bnj1136  30319