Theorem bnj1413 35028

Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1413.1	⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj1413	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ∈ V)

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

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem bnj1413

Step	Hyp	Ref	Expression
1		bnj1148 34989	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
2		bnj893 34921	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
3		simp1 1135	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
4		bnj1127 34984	. . . . . . . 8 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑦 ∈ 𝐴)
5	4	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
6		bnj893 34921	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
7	3, 5, 6	syl2anc 584	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑦, 𝐴, 𝑅) ∈ V)
8	7	3expia 1120	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → trCl(𝑦, 𝐴, 𝑅) ∈ V))
9	8	ralrimiv 3143	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
10		iunexg 7987	. . . 4 ⊢ (( 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)
11	2, 9, 10	syl2anc 584	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ∈ V)
12	1, 11	bnj1149 34785	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ∈ V)
13		bnj1413.1	. . 3 ⊢ 𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
14		bnj906 34923	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
15		iunss1 5011	. . . 4 ⊢ ( pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
16		unss2 4197	. . . 4 ⊢ (∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ⊆ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) → ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
17	14, 15, 16	3syl 18	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
18	13, 17	eqsstrid 4044	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ⊆ ( pred(𝑋, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
19	12, 18	ssexd 5330	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963  ∪ ciun 4996   predc-bnj14 34681   FrSe w-bnj15 34685   trClc-bnj18 34687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-bnj17 34680  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686  df-bnj18 34688

This theorem is referenced by:  bnj1408  35029