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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 35843 | . 2 ⊢ (𝑥 = ∅ → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃))) | |
2 | fveleq 35843 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝑦) ∈ 𝑃))) | |
3 | fveleq 35843 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃))) | |
4 | fveleq 35843 | . 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 7885 | 1 ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∅c0 4317 suc csuc 6359 ‘cfv 6536 ωcom 7851 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fv 6544 df-om 7852 |
This theorem is referenced by: findreccl 35845 |
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