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Theorem bnj554 32879
Description: Technical lemma for bnj852 32901. 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 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj554.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj554.21 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.22 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj554.23 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.24 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
Assertion
Ref Expression
bnj554 ((𝜂𝜁) → ((𝐺𝑚) = 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
StepHypRef Expression
1 bnj554.19 . . 3 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
21bnj1254 32789 . 2 (𝜂𝑚 = suc 𝑝)
3 bnj554.20 . . 3 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
43simp3bi 1146 . 2 (𝜁𝑚 = suc 𝑖)
5 simpr 485 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑚 = suc 𝑖)
6 bnj551 32722 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
7 fveq2 6774 . . . 4 (𝑚 = suc 𝑖 → (𝐺𝑚) = (𝐺‘suc 𝑖))
8 fveq2 6774 . . . . 5 (𝑝 = 𝑖 → (𝐺𝑝) = (𝐺𝑖))
9 iuneq1 4940 . . . . . 6 ((𝐺𝑝) = (𝐺𝑖) → 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
10 bnj554.24 . . . . . 6 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
11 bnj554.23 . . . . . 6 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
129, 10, 113eqtr4g 2803 . . . . 5 ((𝐺𝑝) = (𝐺𝑖) → 𝐿 = 𝐾)
138, 12syl 17 . . . 4 (𝑝 = 𝑖𝐿 = 𝐾)
147, 13eqeqan12d 2752 . . 3 ((𝑚 = suc 𝑖𝑝 = 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
155, 6, 14syl2anc 584 . 2 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
162, 4, 15syl2an 596 1 ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106   ciun 4924  suc csuc 6268  cfv 6433  ωcom 7712  w-bnj17 32665   predc-bnj14 32667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-reg 9351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-eprel 5495  df-fr 5544  df-suc 6272  df-iota 6391  df-fv 6441  df-bnj17 32666
This theorem is referenced by:  bnj558  32882
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