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Theorem bnj1000 32270
 Description: Technical lemma for bnj852 32250. 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
bnj1000.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1000.2 (𝜓″[𝐺 / 𝑓]𝜓)
bnj1000.3 𝐺 ∈ V
bnj1000.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1000.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj1000 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑓   𝑖,𝐺   𝑓,𝑁   𝑅,𝑓   𝑓,𝑖,𝑦   𝑦,𝑛
Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑦,𝑖,𝑚,𝑛)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛)   𝑅(𝑦,𝑖,𝑚,𝑛)   𝐺(𝑦,𝑓,𝑚,𝑛)   𝑁(𝑦,𝑖,𝑚,𝑛)   𝜓″(𝑦,𝑓,𝑖,𝑚,𝑛)

Proof of Theorem bnj1000
Dummy variables 𝑒 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1000.2 . 2 (𝜓″[𝐺 / 𝑓]𝜓)
2 df-ral 3138 . . . . 5 (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
32bicomi 227 . . . 4 (∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
43sbcbii 3814 . . 3 ([𝐺 / 𝑓]𝑖(𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝐺 / 𝑓]𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5 bnj1000.3 . . . . . . 7 𝐺 ∈ V
6 nfv 1916 . . . . . . . 8 𝑓 𝑖 ∈ ω
76sbc19.21g 3830 . . . . . . 7 (𝐺 ∈ V → ([𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → [𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
85, 7ax-mp 5 . . . . . 6 ([𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → [𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
9 nfv 1916 . . . . . . . . . 10 𝑓 suc 𝑖𝑁
109sbc19.21g 3830 . . . . . . . . 9 (𝐺 ∈ V → ([𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
115, 10ax-mp 5 . . . . . . . 8 ([𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
12 fveq1 6660 . . . . . . . . . . 11 (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
13 fveq1 6660 . . . . . . . . . . . 12 (𝑓 = 𝐺 → (𝑓𝑖) = (𝐺𝑖))
14 ax-5 1912 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑓𝑖) → ∀𝑦 𝑤 ∈ (𝑓𝑖))
15 bnj1000.16 . . . . . . . . . . . . . . . 16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
16 nfcv 2982 . . . . . . . . . . . . . . . . 17 𝑦𝑓
17 nfcv 2982 . . . . . . . . . . . . . . . . . . 19 𝑦𝑛
18 bnj1000.15 . . . . . . . . . . . . . . . . . . . 20 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
19 nfiu1 4939 . . . . . . . . . . . . . . . . . . . 20 𝑦 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
2018, 19nfcxfr 2980 . . . . . . . . . . . . . . . . . . 19 𝑦𝐶
2117, 20nfop 4805 . . . . . . . . . . . . . . . . . 18 𝑦𝑛, 𝐶
2221nfsn 4628 . . . . . . . . . . . . . . . . 17 𝑦{⟨𝑛, 𝐶⟩}
2316, 22nfun 4127 . . . . . . . . . . . . . . . 16 𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
2415, 23nfcxfr 2980 . . . . . . . . . . . . . . 15 𝑦𝐺
25 nfcv 2982 . . . . . . . . . . . . . . 15 𝑦𝑖
2624, 25nffv 6671 . . . . . . . . . . . . . 14 𝑦(𝐺𝑖)
2726nfcrii 2974 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐺𝑖) → ∀𝑦 𝑤 ∈ (𝐺𝑖))
2814, 27bnj1316 32149 . . . . . . . . . . . 12 ((𝑓𝑖) = (𝐺𝑖) → 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
2913, 28syl 17 . . . . . . . . . . 11 (𝑓 = 𝐺 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
3012, 29eqeq12d 2840 . . . . . . . . . 10 (𝑓 = 𝐺 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
31 fveq1 6660 . . . . . . . . . . 11 (𝑓 = 𝑒 → (𝑓‘suc 𝑖) = (𝑒‘suc 𝑖))
32 fveq1 6660 . . . . . . . . . . . 12 (𝑓 = 𝑒 → (𝑓𝑖) = (𝑒𝑖))
33 ax-5 1912 . . . . . . . . . . . . 13 ((𝑓𝑖) = (𝑒𝑖) → ∀𝑦(𝑓𝑖) = (𝑒𝑖))
3433bnj956 32105 . . . . . . . . . . . 12 ((𝑓𝑖) = (𝑒𝑖) → 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑒𝑖) pred(𝑦, 𝐴, 𝑅))
3532, 34syl 17 . . . . . . . . . . 11 (𝑓 = 𝑒 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑒𝑖) pred(𝑦, 𝐴, 𝑅))
3631, 35eqeq12d 2840 . . . . . . . . . 10 (𝑓 = 𝑒 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑒‘suc 𝑖) = 𝑦 ∈ (𝑒𝑖) pred(𝑦, 𝐴, 𝑅)))
37 fveq1 6660 . . . . . . . . . . 11 (𝑒 = 𝐺 → (𝑒‘suc 𝑖) = (𝐺‘suc 𝑖))
38 fveq1 6660 . . . . . . . . . . . 12 (𝑒 = 𝐺 → (𝑒𝑖) = (𝐺𝑖))
39 ax-5 1912 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑒𝑖) → ∀𝑦 𝑤 ∈ (𝑒𝑖))
4039, 27bnj1316 32149 . . . . . . . . . . . 12 ((𝑒𝑖) = (𝐺𝑖) → 𝑦 ∈ (𝑒𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4138, 40syl 17 . . . . . . . . . . 11 (𝑒 = 𝐺 𝑦 ∈ (𝑒𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4237, 41eqeq12d 2840 . . . . . . . . . 10 (𝑒 = 𝐺 → ((𝑒‘suc 𝑖) = 𝑦 ∈ (𝑒𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
435, 30, 36, 42bnj610 32075 . . . . . . . . 9 ([𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4443imbi2i 339 . . . . . . . 8 ((suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
4511, 44bitri 278 . . . . . . 7 ([𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
4645imbi2i 339 . . . . . 6 ((𝑖 ∈ ω → [𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
478, 46bitri 278 . . . . 5 ([𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
4847albii 1821 . . . 4 (∀𝑖[𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
49 sbcal 3818 . . . 4 ([𝐺 / 𝑓]𝑖(𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖[𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
50 df-ral 3138 . . . 4 (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
5148, 49, 503bitr4ri 307 . . 3 (∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝐺 / 𝑓]𝑖(𝑖 ∈ ω → (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
52 bnj1000.1 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5352sbcbii 3814 . . 3 ([𝐺 / 𝑓]𝜓[𝐺 / 𝑓]𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
544, 51, 533bitr4ri 307 . 2 ([𝐺 / 𝑓]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
551, 54bitri 278 1 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480  [wsbc 3758   ∪ cun 3917  {csn 4550  ⟨cop 4556  ∪ ciun 4905  suc csuc 6180  ‘cfv 6343  ωcom 7574   predc-bnj14 32015 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-iota 6302  df-fv 6351 This theorem is referenced by:  bnj965  32271
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