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Mirrors > Home > MPE Home > Th. List > Mathboxes > riotaclbgBAD | Structured version Visualization version GIF version |
Description: Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
riotaclbgBAD | ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotacl 7110 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
2 | undefnel2 7926 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ (Undef‘𝐴) ∈ 𝐴) | |
3 | iffalse 4434 | . . . . . . 7 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) | |
4 | ax-riotaBAD 36249 | . . . . . . 7 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) | |
5 | abid1 2931 | . . . . . . . 8 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
6 | 5 | fveq2i 6648 | . . . . . . 7 ⊢ (Undef‘𝐴) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}) |
7 | 3, 4, 6 | 3eqtr4g 2858 | . . . . . 6 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = (Undef‘𝐴)) |
8 | 7 | eleq1d 2874 | . . . . 5 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ (Undef‘𝐴) ∈ 𝐴)) |
9 | 8 | notbid 321 | . . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ¬ (Undef‘𝐴) ∈ 𝐴)) |
10 | 2, 9 | syl5ibrcom 250 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
11 | 10 | con4d 115 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → ∃!𝑥 ∈ 𝐴 𝜑)) |
12 | 1, 11 | impbid2 229 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 {cab 2776 ∃!wreu 3108 ifcif 4425 ℩cio 6281 ‘cfv 6324 ℩crio 7092 Undefcund 7921 |
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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
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-ral 3111 df-rex 3112 df-reu 3113 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-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-undef 7922 |
This theorem is referenced by: riotaclbBAD 36251 riotasvd 36252 |
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