Theorem findfvcl 36646

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 36645	. 2 ⊢ (𝑥 = ∅ → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃)))
2		fveleq 36645	. 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝑦) ∈ 𝑃)))
3		fveleq 36645	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃)))
4		fveleq 36645	. 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 7838	1 ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃))

Syntax hints:   → wi 4   ∈ wcel 2113  ∅c0 4285  suc csuc 6319  ‘cfv 6492  ωcom 7808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fv 6500  df-om 7809

This theorem is referenced by:  findreccl  36647