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Theorem findfvcl 35969
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 35968 . 2 (𝑥 = ∅ → ((𝜑 → (𝐹𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃)))
2 fveleq 35968 . 2 (𝑥 = 𝑦 → ((𝜑 → (𝐹𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹𝑦) ∈ 𝑃)))
3 fveleq 35968 . 2 (𝑥 = suc 𝑦 → ((𝜑 → (𝐹𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃)))
4 fveleq 35968 . 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 7910 1 (𝐴 ∈ ω → (𝜑 → (𝐹𝐴) ∈ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  c0 4326  suc csuc 6376  cfv 6553  ωcom 7876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fv 6561  df-om 7877
This theorem is referenced by:  findreccl  35970
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