Theorem bnj1133 31362

Description: Technical lemma for bnj69 31383. 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 31350	. . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
4	1, 3	bnj769 31137	. 2 ⊢ (𝜒 → E Fr 𝑛)
5		impexp 461	. . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
6	5	bicomi 214	. . . . 5 ⊢ ((𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
7	6	albii 1894	. . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
8		bnj1133.8	. . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)
9	7, 8	mpgbir 1873	. . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))
10		df-ral 3053	. . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
11	9, 10	mpbir 221	. 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)
12		vex 3341	. . 3 ⊢ 𝑛 ∈ V
13		bnj1133.7	. . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
14	12, 13	bnj110 31233	. 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃)
15	4, 11, 14	sylancl 697	1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1628   = wceq 1630   ∈ wcel 2137  ∀wral 3048  [wsbc 3574   ∖ cdif 3710  ∅c0 4056  {csn 4319   class class class wbr 4802   E cep 5176   Fr wfr 5220   Fn wfn 6042  ωcom 7228   ∧ w-bnj17 31059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-tr 4903  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-om 7229  df-bnj17 31060

This theorem is referenced by:  bnj1128  31363