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Theorem bnj1123 31828
Description: Technical lemma for bnj69 31852. 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 3752 . 2 ([𝑗 / 𝑖]𝜂[𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
4 bnj1123.3 . . . . . . . 8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
5 nfcv 2947 . . . . . . . . . 10 𝑖𝐷
6 nfv 1890 . . . . . . . . . . 11 𝑖 𝑓 Fn 𝑛
7 nfv 1890 . . . . . . . . . . 11 𝑖𝜑
8 bnj1123.4 . . . . . . . . . . . . 13 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
98bnj1095 31626 . . . . . . . . . . . 12 (𝜓 → ∀𝑖𝜓)
109nf5i 2115 . . . . . . . . . . 11 𝑖𝜓
116, 7, 10nf3an 1881 . . . . . . . . . 10 𝑖(𝑓 Fn 𝑛𝜑𝜓)
125, 11nfrex 3268 . . . . . . . . 9 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
1312nfab 2953 . . . . . . . 8 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
144, 13nfcxfr 2945 . . . . . . 7 𝑖𝐾
1514nfcri 2941 . . . . . 6 𝑖 𝑓𝐾
16 nfv 1890 . . . . . 6 𝑖 𝑗 ∈ dom 𝑓
1715, 16nfan 1879 . . . . 5 𝑖(𝑓𝐾𝑗 ∈ dom 𝑓)
18 nfv 1890 . . . . 5 𝑖(𝑓𝑗) ⊆ 𝐵
1917, 18nfim 1876 . . . 4 𝑖((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)
20 eleq1w 2863 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓𝑗 ∈ dom 𝑓))
2120anbi2d 628 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝐾𝑖 ∈ dom 𝑓) ↔ (𝑓𝐾𝑗 ∈ dom 𝑓)))
22 fveq2 6530 . . . . . 6 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
2322sseq1d 3914 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝑖) ⊆ 𝐵 ↔ (𝑓𝑗) ⊆ 𝐵))
2421, 23imbi12d 346 . . . 4 (𝑖 = 𝑗 → (((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
2519, 24sbciegf 3733 . . 3 (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
2625elv 3437 . 2 ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
271, 3, 263bitri 298 1 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1078   = wceq 1520  wcel 2079  {cab 2773  wral 3103  wrex 3104  Vcvv 3432  [wsbc 3701  wss 3854   ciun 4819  dom cdm 5435  suc csuc 6060   Fn wfn 6212  cfv 6217  ωcom 7427   predc-bnj14 31531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-iota 6181  df-fv 6225
This theorem is referenced by:  bnj1030  31829
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