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Mathbox for Emmett Weisz |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0setrec | Structured version Visualization version GIF version |
Description: If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.) |
Ref | Expression |
---|---|
0setrec.1 | ⊢ (𝜑 → (𝐹‘∅) = ∅) |
Ref | Expression |
---|---|
0setrec | ⊢ (𝜑 → setrecs(𝐹) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | ss0 4306 | . . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
3 | fveq2 6645 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅)) | |
4 | 0setrec.1 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅) | |
5 | 3, 4 | sylan9eqr 2855 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅) |
6 | 5 | ex 416 | . . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅)) |
7 | eqimss 3971 | . . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅) | |
8 | 2, 6, 7 | syl56 36 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
9 | 8 | alrimiv 1928 | . . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
10 | 1, 9 | setrec2v 45226 | . 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅) |
11 | ss0 4306 | . 2 ⊢ (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅) | |
12 | 10, 11 | syl 17 | 1 ⊢ (𝜑 → setrecs(𝐹) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ⊆ wss 3881 ∅c0 4243 ‘cfv 6324 setrecscsetrecs 45213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-setrecs 45214 |
This theorem is referenced by: (None) |
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