Theorem bnj1136 32977

Description: Technical lemma for bnj69 32990. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1136.1	⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
bnj1136.2	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
bnj1136.3	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))

Assertion

Ref	Expression
bnj1136	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)

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

Allowed substitution hints:   𝜃(𝑦)   𝜏(𝑦)   𝐵(𝑦)

Proof of Theorem bnj1136

Step	Hyp	Ref	Expression
1		bnj1136.2	. . . 4 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
2	1	biimpri 227	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝜃)
3		bnj1136.1	. . . . 5 ⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
4		bnj1148 32976	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
5		bnj893 32908	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
6		simp1 1135	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
7		bnj1127 32971	. . . . . . . . . . 11 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑦 ∈ 𝐴)
8	7	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
9		bnj893 32908	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
10	6, 8, 9	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
11	10	3expia 1120	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ∈ V))
12	11	ralrimiv 3102	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
13		iunexg 7806	. . . . . . 7 ⊢ (( trCl(𝑋, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
14	5, 12, 13	syl2anc 584	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
15	4, 14	bnj1149 32772	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ∈ V)
16	3, 15	eqeltrid 2843	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ∈ V)
17	3	bnj1137 32975	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo(𝐵, 𝐴, 𝑅))
18	3	bnj931 32750	. . . . 5 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵
19	18	a1i 11	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
20		bnj1136.3	. . . 4 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
21	16, 17, 19, 20	syl3anbrc 1342	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝜏)
22	1, 20	bnj1124 32968	. . 3 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
23	2, 21, 22	syl2anc 584	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
24		bnj906 32910	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25		bnj1125 32972	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
26	25	3expia 1120	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
27	26	ralrimiv 3102	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
28		ss2iun 4942	. . . . . 6 ⊢ (∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑋, 𝐴, 𝑅))
29		bnj1143 32770	. . . . . 6 ⊢ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)
30	28, 29	sstrdi 3933	. . . . 5 ⊢ (∀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))
31	27, 30	syl 17	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
32	24, 31	unssd 4120	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ⊆ trCl(𝑋, 𝐴, 𝑅))
33	3, 32	eqsstrid 3969	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ⊆ trCl(𝑋, 𝐴, 𝑅))
34	23, 33	eqssd 3938	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∪ ciun 4924   predc-bnj14 32667   FrSe w-bnj15 32671   trClc-bnj18 32673   TrFow-bnj19 32675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-bnj17 32666  df-bnj14 32668  df-bnj13 32670  df-bnj15 32672  df-bnj18 32674  df-bnj19 32676

This theorem is referenced by:  bnj1408  33016