Theorem bnj554 32281

Description: Technical lemma for bnj852 32303. 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 32191	. 2 ⊢ (𝜂 → 𝑚 = suc 𝑝)
3		bnj554.20	. . 3 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
4	3	simp3bi 1144	. 2 ⊢ (𝜁 → 𝑚 = suc 𝑖)
5		simpr 488	. . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑚 = suc 𝑖)
6		bnj551 32123	. . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖)
7		fveq2 6645	. . . 4 ⊢ (𝑚 = suc 𝑖 → (𝐺‘𝑚) = (𝐺‘suc 𝑖))
8		fveq2 6645	. . . . 5 ⊢ (𝑝 = 𝑖 → (𝐺‘𝑝) = (𝐺‘𝑖))
9		iuneq1 4897	. . . . . 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 2858	. . . . 5 ⊢ ((𝐺‘𝑝) = (𝐺‘𝑖) → 𝐿 = 𝐾)
13	8, 12	syl 17	. . . 4 ⊢ (𝑝 = 𝑖 → 𝐿 = 𝐾)
14	7, 13	eqeqan12d 2815	. . 3 ⊢ ((𝑚 = suc 𝑖 ∧ 𝑝 = 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
15	5, 6, 14	syl2anc 587	. 2 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
16	2, 4, 15	syl2an 598	1 ⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∪ ciun 4881  suc csuc 6161  ‘cfv 6324  ωcom 7560   ∧ w-bnj17 32066   predc-bnj14 32068

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  ax-reg 9040

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  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-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-eprel 5430  df-fr 5478  df-suc 6165  df-iota 6283  df-fv 6332  df-bnj17 32067

This theorem is referenced by:  bnj558  32284