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Theorem bnj1123 31371
Description: Technical lemma for bnj69 31395. 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 3683 . 2 ([𝑗 / 𝑖]𝜂[𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
4 vex 3390 . . 3 𝑗 ∈ V
5 bnj1123.3 . . . . . . . 8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
6 nfcv 2944 . . . . . . . . . 10 𝑖𝐷
7 nfv 2005 . . . . . . . . . . 11 𝑖 𝑓 Fn 𝑛
8 nfv 2005 . . . . . . . . . . 11 𝑖𝜑
9 bnj1123.4 . . . . . . . . . . . . 13 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
109bnj1095 31169 . . . . . . . . . . . 12 (𝜓 → ∀𝑖𝜓)
1110nf5i 2189 . . . . . . . . . . 11 𝑖𝜓
127, 8, 11nf3an 1993 . . . . . . . . . 10 𝑖(𝑓 Fn 𝑛𝜑𝜓)
136, 12nfrex 3190 . . . . . . . . 9 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
1413nfab 2949 . . . . . . . 8 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
155, 14nfcxfr 2942 . . . . . . 7 𝑖𝐾
1615nfcri 2938 . . . . . 6 𝑖 𝑓𝐾
17 nfv 2005 . . . . . 6 𝑖 𝑗 ∈ dom 𝑓
1816, 17nfan 1990 . . . . 5 𝑖(𝑓𝐾𝑗 ∈ dom 𝑓)
19 nfv 2005 . . . . 5 𝑖(𝑓𝑗) ⊆ 𝐵
2018, 19nfim 1987 . . . 4 𝑖((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)
21 eleq1w 2864 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓𝑗 ∈ dom 𝑓))
2221anbi2d 616 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝐾𝑖 ∈ dom 𝑓) ↔ (𝑓𝐾𝑗 ∈ dom 𝑓)))
23 fveq2 6402 . . . . . 6 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
2423sseq1d 3823 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝑖) ⊆ 𝐵 ↔ (𝑓𝑗) ⊆ 𝐵))
2522, 24imbi12d 335 . . . 4 (𝑖 = 𝑗 → (((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
2620, 25sbciegf 3659 . . 3 (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
274, 26ax-mp 5 . 2 ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
281, 3, 273bitri 288 1 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2155  {cab 2788  wral 3092  wrex 3093  Vcvv 3387  [wsbc 3627  wss 3763   ciun 4705  dom cdm 5305  suc csuc 5932   Fn wfn 6090  cfv 6095  ωcom 7289   predc-bnj14 31073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-iota 6058  df-fv 6103
This theorem is referenced by:  bnj1030  31372
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