Theorem bnj554 35319

Description: Technical lemma for bnj852 35341. 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
bnj554.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj554.20	⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
bnj554.21	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.22	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj554.23	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.24	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)

Assertion

Ref	Expression
bnj554	⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))

Distinct variable groups:   𝑦,𝐺   𝑦,𝑖   𝑦,𝑝

Allowed substitution hints:   𝜂(𝑦, 𝑖, 𝑚, 𝑛, 𝑝)   𝜁(𝑦, 𝑖, 𝑚, 𝑛, 𝑝)   𝐴(𝑦, 𝑖, 𝑚, 𝑛, 𝑝)   𝐷(𝑦, 𝑖, 𝑚, 𝑛, 𝑝)   𝑅(𝑦, 𝑖, 𝑚, 𝑛, 𝑝)   𝐺(𝑖, 𝑚, 𝑛, 𝑝)   𝐾(𝑦, 𝑖, 𝑚, 𝑛, 𝑝)   𝐿(𝑦, 𝑖, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj554

Step	Hyp	Ref	Expression
1		bnj554.19	. . 3 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2	1	bnj1254 35229	. 2 ⊢ (𝜂 → 𝑚 = suc 𝑝)
3		bnj554.20	. . 3 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
4	3	simp3bi 1165	. 2 ⊢ (𝜁 → 𝑚 = suc 𝑖)
5		simpr 490	. . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑚 = suc 𝑖)
6		bnj551 35163	. . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖)
7		fveq2 6888	. . . 4 ⊢ (𝑚 = suc 𝑖 → (𝐺‘𝑚) = (𝐺‘suc 𝑖))
8		fveq2 6888	. . . . 5 ⊢ (𝑝 = 𝑖 → (𝐺‘𝑝) = (𝐺‘𝑖))
9		iuneq1 4978	. . . . . 6 ⊢ ((𝐺‘𝑝) = (𝐺‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
10		bnj554.24	. . . . . 6 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
11		bnj554.23	. . . . . 6 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
12	9, 10, 11	3eqtr4g 2826	. . . . 5 ⊢ ((𝐺‘𝑝) = (𝐺‘𝑖) → 𝐿 = 𝐾)
13	8, 12	syl 18	. . . 4 ⊢ (𝑝 = 𝑖 → 𝐿 = 𝐾)
14	7, 13	eqeqan12d 2780	. . 3 ⊢ ((𝑚 = suc 𝑖 ∧ 𝑝 = 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
15	5, 6, 14	syl2anc 596	. 2 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
16	2, 4, 15	syl2an 608	1 ⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∪ ciun 4961  suc csuc 6369  ‘cfv 6543  ωcom 7871   ∧ w-bnj17 35107   predc-bnj14 35109

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745  ax-reg 9564

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-eprel 5566  df-fr 5619  df-suc 6373  df-iota 6499  df-fv 6551  df-bnj17 35108

This theorem is used by:  bnj558  35322