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Theorem bnj106 31245
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj106.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj106.2 𝐹 ∈ V
Assertion
Ref Expression
bnj106 ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑓,𝑛   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑛   𝑖,𝑛,𝑦
Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐹(𝑛)

Proof of Theorem bnj106
StepHypRef Expression
1 bnj106.1 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2 bnj105 31099 . . . 4 1𝑜 ∈ V
31, 2bnj92 31239 . . 3 ([1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
43sbcbii 3632 . 2 ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜓[𝐹 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5 bnj106.2 . . 3 𝐹 ∈ V
6 fveq1 6351 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘suc 𝑖) = (𝐹‘suc 𝑖))
7 fveq1 6351 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑖) = (𝐹𝑖))
87bnj1113 31163 . . . . . 6 (𝑓 = 𝐹 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
96, 8eqeq12d 2775 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
109imbi2d 329 . . . 4 (𝑓 = 𝐹 → ((suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))))
1110ralbidv 3124 . . 3 (𝑓 = 𝐹 → (∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))))
125, 11sbcie 3611 . 2 ([𝐹 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
134, 12bitri 264 1 ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  [wsbc 3576   ciun 4672  suc csuc 5886  cfv 6049  ωcom 7230  1𝑜c1o 7722   predc-bnj14 31063
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 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992
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 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-uni 4589  df-iun 4674  df-br 4805  df-suc 5890  df-iota 6012  df-fv 6057  df-1o 7729
This theorem is referenced by:  bnj126  31250
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