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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 2726 | . . 3 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | ss0 4393 | . . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
3 | fveq2 6884 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅)) | |
4 | 0setrec.1 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅) | |
5 | 3, 4 | sylan9eqr 2788 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅) |
6 | 5 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅)) |
7 | eqimss 4035 | . . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅) | |
8 | 2, 6, 7 | syl56 36 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
9 | 8 | alrimiv 1922 | . . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
10 | 1, 9 | setrec2v 47997 | . 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅) |
11 | ss0 4393 | . 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 1533 ⊆ wss 3943 ∅c0 4317 ‘cfv 6536 setrecscsetrecs 47984 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 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-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-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fv 6544 df-setrecs 47985 |
This theorem is referenced by: (None) |
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