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Theorem bnj155 31466
Description: Technical lemma for bnj153 31467. 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
bnj155.1 (𝜓1[𝑔 / 𝑓]𝜓′)
bnj155.2 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj155 (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑔,𝑖,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑔,𝑖)   𝑅(𝑦,𝑔,𝑖)   𝜓′(𝑦,𝑓,𝑔,𝑖)   𝜓1(𝑦,𝑓,𝑔,𝑖)

Proof of Theorem bnj155
StepHypRef Expression
1 bnj155.1 . 2 (𝜓1[𝑔 / 𝑓]𝜓′)
2 bnj155.2 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
32sbcbii 3689 . 2 ([𝑔 / 𝑓]𝜓′[𝑔 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 vex 3388 . . 3 𝑔 ∈ V
5 fveq1 6410 . . . . . 6 (𝑓 = 𝑔 → (𝑓‘suc 𝑖) = (𝑔‘suc 𝑖))
6 fveq1 6410 . . . . . . 7 (𝑓 = 𝑔 → (𝑓𝑖) = (𝑔𝑖))
76iuneq1d 4735 . . . . . 6 (𝑓 = 𝑔 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))
85, 7eqeq12d 2814 . . . . 5 (𝑓 = 𝑔 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
98imbi2d 332 . . . 4 (𝑓 = 𝑔 → ((suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
109ralbidv 3167 . . 3 (𝑓 = 𝑔 → (∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
114, 10sbcie 3668 . 2 ([𝑔 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
121, 3, 113bitri 289 1 (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  wral 3089  [wsbc 3633   ciun 4710  suc csuc 5943  cfv 6101  ωcom 7299  1𝑜c1o 7792   predc-bnj14 31274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-sbc 3634  df-in 3776  df-ss 3783  df-uni 4629  df-iun 4712  df-br 4844  df-iota 6064  df-fv 6109
This theorem is referenced by:  bnj153  31467
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