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Theorem bnj554 32442
Description: Technical lemma for bnj852 32464. 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 32352 . 2 (𝜂𝑚 = suc 𝑝)
3 bnj554.20 . . 3 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
43simp3bi 1148 . 2 (𝜁𝑚 = suc 𝑖)
5 simpr 488 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑚 = suc 𝑖)
6 bnj551 32284 . . 3 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
7 fveq2 6668 . . . 4 (𝑚 = suc 𝑖 → (𝐺𝑚) = (𝐺‘suc 𝑖))
8 fveq2 6668 . . . . 5 (𝑝 = 𝑖 → (𝐺𝑝) = (𝐺𝑖))
9 iuneq1 4894 . . . . . 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 2798 . . . . 5 ((𝐺𝑝) = (𝐺𝑖) → 𝐿 = 𝐾)
138, 12syl 17 . . . 4 (𝑝 = 𝑖𝐿 = 𝐾)
147, 13eqeqan12d 2755 . . 3 ((𝑚 = suc 𝑖𝑝 = 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
155, 6, 14syl2anc 587 . 2 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
162, 4, 15syl2an 599 1 ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113   ciun 4878  suc csuc 6168  cfv 6333  ωcom 7593  w-bnj17 32227   predc-bnj14 32229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473  ax-reg 9122
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-eprel 5430  df-fr 5478  df-suc 6172  df-iota 6291  df-fv 6341  df-bnj17 32228
This theorem is referenced by:  bnj558  32445
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