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Mirrors > Home > MPE Home > Th. List > riota5 | Structured version Visualization version GIF version |
Description: A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
Ref | Expression |
---|---|
riota5.1 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
riota5.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵)) |
Ref | Expression |
---|---|
riota5 | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcvd 2893 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
2 | riota5.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
3 | riota5.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵)) | |
4 | 1, 2, 3 | riota5f 7409 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ℩crio 7379 |
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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 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 3365 df-v 3464 df-sbc 3777 df-un 3952 df-ss 3964 df-sn 4634 df-pr 4636 df-uni 4914 df-iota 6506 df-riota 7380 |
This theorem is referenced by: f1ocnvfv3 7419 ttrcltr 9759 sqrt0 15246 lubid 18387 lubun 18540 odval2 19549 adjvalval 31870 xdivpnfrp 32794 xrsinvgval 32888 dfgcd3 37031 poimirlem6 37327 poimirlem7 37328 lub0N 38887 glb0N 38891 trlval2 39862 cdlemefrs32fva 40099 cdleme32fva 40136 cdlemg1a 40269 unxpwdom3 42756 |
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