Theorem bnj1133 30311

Description: Technical lemma for bnj69 30332. 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 30299	. . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
4	1, 3	bnj769 30086	. 2 ⊢ (𝜒 → E Fr 𝑛)
5		impexp 461	. . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
6	5	bicomi 213	. . . . 5 ⊢ ((𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
7	6	albii 1737	. . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
8		bnj1133.8	. . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)
9	7, 8	mpgbir 1717	. . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))
10		df-ral 2901	. . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
11	9, 10	mpbir 220	. 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)
12		vex 3176	. . 3 ⊢ 𝑛 ∈ V
13		bnj1133.7	. . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
14	12, 13	bnj110 30182	. 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃)
15	4, 11, 14	sylancl 693	1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  [wsbc 3402   ∖ cdif 3537  ∅c0 3874  {csn 4125   class class class wbr 4583   E cep 4947   Fr wfr 4994   Fn wfn 5799  ωcom 6957   ∧ w-bnj17 30005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958  df-bnj17 30006

This theorem is referenced by:  bnj1128  30312