Theorem bnj1133 32371

Description: Technical lemma for bnj69 32392. 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 32359	. . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
4	1, 3	bnj769 32143	. 2 ⊢ (𝜒 → E Fr 𝑛)
5		impexp 454	. . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
6	5	bicomi 227	. . . . 5 ⊢ ((𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
7	6	albii 1821	. . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
8		bnj1133.8	. . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)
9	7, 8	mpgbir 1801	. . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))
10		df-ral 3111	. . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
11	9, 10	mpbir 234	. 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)
12		vex 3444	. . 3 ⊢ 𝑛 ∈ V
13		bnj1133.7	. . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
14	12, 13	bnj110 32240	. 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃)
15	4, 11, 14	sylancl 589	1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106  [wsbc 3720   ∖ cdif 3878  ∅c0 4243  {csn 4525   class class class wbr 5030   E cep 5429   Fr wfr 5475   Fn wfn 6319  ωcom 7560   ∧ w-bnj17 32066

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-om 7561  df-bnj17 32067

This theorem is referenced by:  bnj1128  32372