Theorem bnj1133 35001

Description: Technical lemma for bnj69 35022. 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
bnj1133.3	⊢ 𝐷 = (ω ∖ {∅})
bnj1133.5	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1133.7	⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
bnj1133.8	⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)

Assertion

Ref	Expression
bnj1133	⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)

Distinct variable groups:   𝑖,𝑗,𝑛   𝜃,𝑗

Allowed substitution hints:   𝜑(𝑓,𝑖,𝑗,𝑛)   𝜓(𝑓,𝑖,𝑗,𝑛)   𝜒(𝑓,𝑖,𝑗,𝑛)   𝜃(𝑓,𝑖,𝑛)   𝜏(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1133

Step	Hyp	Ref	Expression
1		bnj1133.5	. . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj1133.3	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
3	2	bnj1071 34989	. . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
4	1, 3	bnj769 34774	. 2 ⊢ (𝜒 → E Fr 𝑛)
5		impexp 450	. . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
6	5	bicomi 224	. . . . 5 ⊢ ((𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
7	6	albii 1820	. . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
8		bnj1133.8	. . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)
9	7, 8	mpgbir 1800	. . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))
10		df-ral 3048	. . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
11	9, 10	mpbir 231	. 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)
12		vex 3440	. . 3 ⊢ 𝑛 ∈ V
13		bnj1133.7	. . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
14	12, 13	bnj110 34870	. 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃)
15	4, 11, 14	sylancl 586	1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047  [wsbc 3736   ∖ cdif 3894  ∅c0 4280  {csn 4573   class class class wbr 5089   E cep 5513   Fr wfr 5564   Fn wfn 6476  ωcom 7796   ∧ w-bnj17 34698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-tr 5197  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310  df-om 7797  df-bnj17 34699

This theorem is referenced by:  bnj1128  35002