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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 7388 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
2 | undefnel2 8282 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ (Undef‘𝐴) ∈ 𝐴) | |
3 | iffalse 4533 | . . . . . . 7 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) | |
4 | ax-riotaBAD 38662 | . . . . . . 7 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) | |
5 | abid1 2863 | . . . . . . . 8 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
6 | 5 | fveq2i 6894 | . . . . . . 7 ⊢ (Undef‘𝐴) = (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}) |
7 | 3, 4, 6 | 3eqtr4g 2791 | . . . . . 6 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = (Undef‘𝐴)) |
8 | 7 | eleq1d 2811 | . . . . 5 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ (Undef‘𝐴) ∈ 𝐴)) |
9 | 8 | notbid 317 | . . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ¬ (Undef‘𝐴) ∈ 𝐴)) |
10 | 2, 9 | syl5ibrcom 246 | . . 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 225 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 {cab 2703 ∃!wreu 3363 ifcif 4524 ℩cio 6494 ‘cfv 6544 ℩crio 7369 Undefcund 8277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-riotaBAD 38662 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6496 df-fun 6546 df-fv 6552 df-riota 7370 df-undef 8278 |
This theorem is referenced by: riotaclbBAD 38664 riotasvd 38665 |
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