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Mirrors > Home > MPE Home > Th. List > Mathboxes > findfvcl | Structured version Visualization version GIF version |
Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
Ref | Expression |
---|---|
findfvcl.1 | ⊢ (𝜑 → (𝐹‘∅) ∈ 𝑃) |
findfvcl.2 | ⊢ (𝑦 ∈ ω → (𝜑 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃))) |
Ref | Expression |
---|---|
findfvcl | ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveleq 31620 | . 2 ⊢ (𝑥 = ∅ → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃))) | |
2 | fveleq 31620 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝑦) ∈ 𝑃))) | |
3 | fveleq 31620 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃))) | |
4 | fveleq 31620 | . 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 6984 | 1 ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∅c0 3874 suc csuc 5642 ‘cfv 5804 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fv 5812 df-om 6958 |
This theorem is referenced by: findreccl 31622 |
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