Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1123 Structured version   Visualization version   GIF version

Theorem bnj1123 31383
Description: Technical lemma for bnj69 31407. 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 3633 . 2 ([𝑗 / 𝑖]𝜂[𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
4 vex 3344 . . 3 𝑗 ∈ V
5 bnj1123.3 . . . . . . . 8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
6 nfcv 2903 . . . . . . . . . 10 𝑖𝐷
7 nfv 1993 . . . . . . . . . . 11 𝑖 𝑓 Fn 𝑛
8 nfv 1993 . . . . . . . . . . 11 𝑖𝜑
9 bnj1123.4 . . . . . . . . . . . . 13 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
109bnj1095 31181 . . . . . . . . . . . 12 (𝜓 → ∀𝑖𝜓)
1110nf5i 2174 . . . . . . . . . . 11 𝑖𝜓
127, 8, 11nf3an 1981 . . . . . . . . . 10 𝑖(𝑓 Fn 𝑛𝜑𝜓)
136, 12nfrex 3146 . . . . . . . . 9 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
1413nfab 2908 . . . . . . . 8 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
155, 14nfcxfr 2901 . . . . . . 7 𝑖𝐾
1615nfcri 2897 . . . . . 6 𝑖 𝑓𝐾
17 nfv 1993 . . . . . 6 𝑖 𝑗 ∈ dom 𝑓
1816, 17nfan 1978 . . . . 5 𝑖(𝑓𝐾𝑗 ∈ dom 𝑓)
19 nfv 1993 . . . . 5 𝑖(𝑓𝑗) ⊆ 𝐵
2018, 19nfim 1975 . . . 4 𝑖((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)
21 eleq1w 2823 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓𝑗 ∈ dom 𝑓))
2221anbi2d 742 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝐾𝑖 ∈ dom 𝑓) ↔ (𝑓𝐾𝑗 ∈ dom 𝑓)))
23 fveq2 6354 . . . . . 6 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
2423sseq1d 3774 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝑖) ⊆ 𝐵 ↔ (𝑓𝑗) ⊆ 𝐵))
2522, 24imbi12d 333 . . . 4 (𝑖 = 𝑗 → (((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
2620, 25sbciegf 3609 . . 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 286 1 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  {cab 2747  wral 3051  wrex 3052  Vcvv 3341  [wsbc 3577  wss 3716   ciun 4673  dom cdm 5267  suc csuc 5887   Fn wfn 6045  cfv 6050  ωcom 7232   predc-bnj14 31085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-iota 6013  df-fv 6058
This theorem is referenced by:  bnj1030  31384
  Copyright terms: Public domain W3C validator