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Theorem findfvcl 33357
Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
Hypotheses
Ref Expression
findfvcl.1 (𝜑 → (𝐹‘∅) ∈ 𝑃)
findfvcl.2 (𝑦 ∈ ω → (𝜑 → ((𝐹𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))
Assertion
Ref Expression
findfvcl (𝐴 ∈ ω → (𝜑 → (𝐹𝐴) ∈ 𝑃))
Distinct variable groups:   𝑦,𝐹   𝑦,𝑃   𝜑,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem findfvcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveleq 33356 . 2 (𝑥 = ∅ → ((𝜑 → (𝐹𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃)))
2 fveleq 33356 . 2 (𝑥 = 𝑦 → ((𝜑 → (𝐹𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹𝑦) ∈ 𝑃)))
3 fveleq 33356 . 2 (𝑥 = suc 𝑦 → ((𝜑 → (𝐹𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃)))
4 fveleq 33356 . 2 (𝑥 = 𝐴 → ((𝜑 → (𝐹𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐴) ∈ 𝑃)))
5 findfvcl.1 . 2 (𝜑 → (𝐹‘∅) ∈ 𝑃)
6 findfvcl.2 . . 3 (𝑦 ∈ ω → (𝜑 → ((𝐹𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))
76a2d 29 . 2 (𝑦 ∈ ω → ((𝜑 → (𝐹𝑦) ∈ 𝑃) → (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃)))
81, 2, 3, 4, 5, 7finds 7421 1 (𝐴 ∈ ω → (𝜑 → (𝐹𝐴) ∈ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2051  c0 4172  suc csuc 6028  cfv 6185  ωcom 7394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-tr 5027  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fv 6193  df-om 7395
This theorem is referenced by:  findreccl  33358
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