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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 35968 | . 2 ⊢ (𝑥 = ∅ → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘∅) ∈ 𝑃))) | |
2 | fveleq 35968 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝑦) ∈ 𝑃))) | |
3 | fveleq 35968 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜑 → (𝐹‘𝑥) ∈ 𝑃) ↔ (𝜑 → (𝐹‘suc 𝑦) ∈ 𝑃))) | |
4 | fveleq 35968 | . 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 7910 | 1 ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∅c0 4326 suc csuc 6376 ‘cfv 6553 ωcom 7876 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fv 6561 df-om 7877 |
This theorem is referenced by: findreccl 35970 |
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