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Theorem bnj155 30203
Description: Technical lemma for bnj153 30204. 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 3458 . 2 ([𝑔 / 𝑓]𝜓′[𝑔 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 vex 3176 . . 3 𝑔 ∈ V
5 fveq1 6102 . . . . . 6 (𝑓 = 𝑔 → (𝑓‘suc 𝑖) = (𝑔‘suc 𝑖))
6 fveq1 6102 . . . . . . 7 (𝑓 = 𝑔 → (𝑓𝑖) = (𝑔𝑖))
76iuneq1d 4481 . . . . . 6 (𝑓 = 𝑔 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))
85, 7eqeq12d 2625 . . . . 5 (𝑓 = 𝑔 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
98imbi2d 329 . . . 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 2969 . . 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 3437 . 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 285 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 195   = wceq 1475  wcel 1977  wral 2896  [wsbc 3402   ciun 4455  suc csuc 5642  cfv 5804  ωcom 6957  1𝑜c1o 7440   predc-bnj14 30007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-in 3547  df-ss 3554  df-uni 4373  df-iun 4457  df-br 4584  df-iota 5768  df-fv 5812
This theorem is referenced by:  bnj153  30204
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