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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 2956 | . 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 7121 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ℩crio 7092 |
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 |
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-v 3443 df-sbc 3721 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 df-riota 7093 |
This theorem is referenced by: f1ocnvfv3 7131 sqrt0 14593 lubid 17592 lubun 17725 odval2 18671 adjvalval 29720 xdivpnfrp 30635 xrsinvgval 30711 dfgcd3 34738 poimirlem6 35063 poimirlem7 35064 lub0N 36485 glb0N 36489 trlval2 37459 cdlemefrs32fva 37696 cdleme32fva 37733 cdlemg1a 37866 unxpwdom3 40039 |
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