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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 2610 | . . 3 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | ss0 3926 | . . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
3 | fveq2 6103 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅)) | |
4 | 0setrec.1 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅) | |
5 | 3, 4 | sylan9eqr 2666 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅) |
6 | 5 | ex 449 | . . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅)) |
7 | eqimss 3620 | . . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅) | |
8 | 2, 6, 7 | syl56 35 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
9 | 8 | alrimiv 1842 | . . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
10 | 1, 9 | setrec2v 42242 | . 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅) |
11 | ss0 3926 | . 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 1475 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 setrecscsetrecs 42229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-setrecs 42230 |
This theorem is referenced by: (None) |
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