![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > riota2 | Structured version Visualization version GIF version |
Description: This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
riota2.1 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
riota2 | ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐵 | |
2 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | riota2.1 | . 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | riota2f 6532 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃!wreu 2898 ℩crio 6510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-v 3175 df-sbc 3403 df-un 3545 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-riota 6511 |
This theorem is referenced by: eqsup 8245 sup0 8255 fin23lem22 9032 subadd 10163 divmul 10567 fllelt 12460 flflp1 12470 flval2 12477 flbi 12479 remim 13705 resqrtcl 13842 resqrtthlem 13843 sqrtneg 13856 sqrtthlem 13950 divalgmod 14967 divalgmodOLD 14968 qnumdenbi 15290 catidd 16164 lubprop 16809 glbprop 16822 isglbd 16940 poslubd 16971 ismgmid 17087 isgrpinv 17295 pj1id 17935 coeeq 23787 ismir 25354 mireq 25360 ismidb 25470 islmib 25479 usgraidx2vlem2 25921 nbgraf1olem3 25972 frgrancvvdeqlem4 26560 cnidOLD 26821 hilid 27402 pjpreeq 27641 cnvbraval 28353 cdj3lem2 28678 xdivmul 28964 cvmliftphtlem 30553 cvmlift3lem4 30558 cvmlift3lem6 30560 cvmlift3lem9 30563 transportprops 31311 ltflcei 32567 cmpidelt 32828 exidresid 32848 lshpkrlem1 33415 cdlemeiota 34891 dochfl1 35783 hgmapvs 36201 wessf1ornlem 38366 fourierdlem50 39049 usgredg2vlem2 40453 frgrncvvdeqlem4 41472 |
Copyright terms: Public domain | W3C validator |