Theorem findfvcl 36665

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 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))

Assertion

Ref	Expression
findfvcl	⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 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.

Step	Hyp	Ref	Expression
1		fveleq 36664	. 2 ⊢ (𝑥 = ∅ → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃)))
2		fveleq 36664	. 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝑦) ∈ 𝑃)))
3		fveleq 36664	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃)))
4		fveleq 36664	. 2 ⊢ (𝑥 = 𝐴 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐴) ∈ 𝑃)))
5		findfvcl.1	. 2 ⊢ (𝜑 → (𝐹‘∅) ∈ 𝑃)
6		findfvcl.2	. . 3 ⊢ (𝑦 ∈ ω → (𝜑 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))
7	6	a2d 29	. 2 ⊢ (𝑦 ∈ ω → ((𝜑 → (𝐹‘𝑦) ∈ 𝑃) → (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃)))
8	1, 2, 3, 4, 5, 7	finds 7848	1 ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃))

Syntax hints:   → wi 4   ∈ wcel 2114  ∅c0 4287  suc csuc 6327  ‘cfv 6500  ωcom 7818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fv 6508  df-om 7819

This theorem is referenced by:  findreccl  36666