Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1000 Structured version   Visualization version   GIF version

Theorem bnj1000 34934
Description: Technical lemma for bnj852 34914. 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 3060 . . . . 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 224 . . . 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 3852 . . 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 1912 . . . . . . . 8 𝑓 𝑖 ∈ ω
76sbc19.21g 3869 . . . . . . 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 1912 . . . . . . . . . 10 𝑓 suc 𝑖𝑁
109sbc19.21g 3869 . . . . . . . . 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 6906 . . . . . . . . . . 11 (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
13 fveq1 6906 . . . . . . . . . . . 12 (𝑓 = 𝐺 → (𝑓𝑖) = (𝐺𝑖))
14 ax-5 1908 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑓𝑖) → ∀𝑦 𝑤 ∈ (𝑓𝑖))
15 bnj1000.16 . . . . . . . . . . . . . . . 16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
16 nfcv 2903 . . . . . . . . . . . . . . . . 17 𝑦𝑓
17 nfcv 2903 . . . . . . . . . . . . . . . . . . 19 𝑦𝑛
18 bnj1000.15 . . . . . . . . . . . . . . . . . . . 20 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
19 nfiu1 5032 . . . . . . . . . . . . . . . . . . . 20 𝑦 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
2018, 19nfcxfr 2901 . . . . . . . . . . . . . . . . . . 19 𝑦𝐶
2117, 20nfop 4894 . . . . . . . . . . . . . . . . . 18 𝑦𝑛, 𝐶
2221nfsn 4712 . . . . . . . . . . . . . . . . 17 𝑦{⟨𝑛, 𝐶⟩}
2316, 22nfun 4180 . . . . . . . . . . . . . . . 16 𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
2415, 23nfcxfr 2901 . . . . . . . . . . . . . . 15 𝑦𝐺
25 nfcv 2903 . . . . . . . . . . . . . . 15 𝑦𝑖
2624, 25nffv 6917 . . . . . . . . . . . . . 14 𝑦(𝐺𝑖)
2726nfcrii 2898 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐺𝑖) → ∀𝑦 𝑤 ∈ (𝐺𝑖))
2814, 27bnj1316 34813 . . . . . . . . . . . 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 2751 . . . . . . . . . 10 (𝑓 = 𝐺 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
31 fveq1 6906 . . . . . . . . . . 11 (𝑓 = 𝑒 → (𝑓‘suc 𝑖) = (𝑒‘suc 𝑖))
32 fveq1 6906 . . . . . . . . . . . 12 (𝑓 = 𝑒 → (𝑓𝑖) = (𝑒𝑖))
33 ax-5 1908 . . . . . . . . . . . . 13 ((𝑓𝑖) = (𝑒𝑖) → ∀𝑦(𝑓𝑖) = (𝑒𝑖))
3433bnj956 34769 . . . . . . . . . . . 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 2751 . . . . . . . . . 10 (𝑓 = 𝑒 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑒‘suc 𝑖) = 𝑦 ∈ (𝑒𝑖) pred(𝑦, 𝐴, 𝑅)))
37 fveq1 6906 . . . . . . . . . . 11 (𝑒 = 𝐺 → (𝑒‘suc 𝑖) = (𝐺‘suc 𝑖))
38 fveq1 6906 . . . . . . . . . . . 12 (𝑒 = 𝐺 → (𝑒𝑖) = (𝐺𝑖))
39 ax-5 1908 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑒𝑖) → ∀𝑦 𝑤 ∈ (𝑒𝑖))
4039, 27bnj1316 34813 . . . . . . . . . . . 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 2751 . . . . . . . . . 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 34740 . . . . . . . . 9 ([𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4443imbi2i 336 . . . . . . . 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 275 . . . . . . 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 336 . . . . . 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 275 . . . . 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 1816 . . . 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 3855 . . . 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 3060 . . . 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 304 . . 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 3852 . . 3 ([𝐺 / 𝑓]𝜓[𝐺 / 𝑓]𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
544, 51, 533bitr4ri 304 . 2 ([𝐺 / 𝑓]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
551, 54bitri 275 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 206  wal 1535   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  [wsbc 3791  cun 3961  {csn 4631  cop 4637   ciun 4996  suc csuc 6388  cfv 6563  ωcom 7887   predc-bnj14 34681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by:  bnj965  34935
  Copyright terms: Public domain W3C validator