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Theorem bnj1123 33655
Description: Technical lemma for bnj69 33679. 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
bnj1123.4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1123.3 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1123.1 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
bnj1123.2 (𝜂′[𝑗 / 𝑖]𝜂)
Assertion
Ref Expression
bnj1123 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Distinct variable groups:   𝐵,𝑖   𝐷,𝑖   𝑓,𝑖   𝑖,𝑗   𝑖,𝑛   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑗,𝑛)   𝑅(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐾(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1123
StepHypRef Expression
1 bnj1123.2 . 2 (𝜂′[𝑗 / 𝑖]𝜂)
2 bnj1123.1 . . 3 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
32sbcbii 3800 . 2 ([𝑗 / 𝑖]𝜂[𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
4 bnj1123.3 . . . . . . . 8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
5 nfcv 2904 . . . . . . . . . 10 𝑖𝐷
6 nfv 1918 . . . . . . . . . . 11 𝑖 𝑓 Fn 𝑛
7 nfv 1918 . . . . . . . . . . 11 𝑖𝜑
8 bnj1123.4 . . . . . . . . . . . . 13 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
98bnj1095 33450 . . . . . . . . . . . 12 (𝜓 → ∀𝑖𝜓)
109nf5i 2143 . . . . . . . . . . 11 𝑖𝜓
116, 7, 10nf3an 1905 . . . . . . . . . 10 𝑖(𝑓 Fn 𝑛𝜑𝜓)
125, 11nfrexw 3295 . . . . . . . . 9 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
1312nfab 2910 . . . . . . . 8 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
144, 13nfcxfr 2902 . . . . . . 7 𝑖𝐾
1514nfcri 2891 . . . . . 6 𝑖 𝑓𝐾
16 nfv 1918 . . . . . 6 𝑖 𝑗 ∈ dom 𝑓
1715, 16nfan 1903 . . . . 5 𝑖(𝑓𝐾𝑗 ∈ dom 𝑓)
18 nfv 1918 . . . . 5 𝑖(𝑓𝑗) ⊆ 𝐵
1917, 18nfim 1900 . . . 4 𝑖((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)
20 eleq1w 2817 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓𝑗 ∈ dom 𝑓))
2120anbi2d 630 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝐾𝑖 ∈ dom 𝑓) ↔ (𝑓𝐾𝑗 ∈ dom 𝑓)))
22 fveq2 6843 . . . . . 6 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
2322sseq1d 3976 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝑖) ⊆ 𝐵 ↔ (𝑓𝑗) ⊆ 𝐵))
2421, 23imbi12d 345 . . . 4 (𝑖 = 𝑗 → (((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
2519, 24sbciegf 3779 . . 3 (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
2625elv 3450 . 2 ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
271, 3, 263bitri 297 1 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  Vcvv 3444  [wsbc 3740  wss 3911   ciun 4955  dom cdm 5634  suc csuc 6320   Fn wfn 6492  cfv 6497  ωcom 7803   predc-bnj14 33357
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505
This theorem is referenced by:  bnj1030  33656
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