![]() |
Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
|
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 2733 | . . 3 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | ss0 4359 | . . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅) | |
3 | fveq2 6843 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅)) | |
4 | 0setrec.1 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅) | |
5 | 3, 4 | sylan9eqr 2795 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅) |
6 | 5 | ex 414 | . . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅)) |
7 | eqimss 4001 | . . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅) | |
8 | 2, 6, 7 | syl56 36 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
9 | 8 | alrimiv 1931 | . . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅)) |
10 | 1, 9 | setrec2v 47227 | . 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅) |
11 | ss0 4359 | . 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 1542 ⊆ wss 3911 ∅c0 4283 ‘cfv 6497 setrecscsetrecs 47214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 df-setrecs 47215 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |