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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 34377 | . 2 ⊢ (𝑥 = ∅ → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃))) | |
2 | fveleq 34377 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝑦) ∈ 𝑃))) | |
3 | fveleq 34377 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃))) | |
4 | fveleq 34377 | . 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 7676 | 1 ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∅c0 4237 suc csuc 6215 ‘cfv 6380 ωcom 7644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fv 6388 df-om 7645 |
This theorem is referenced by: findreccl 34379 |
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