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Mirrors > Home > ILE Home > Th. List > ralrnmpt | Unicode version |
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
ralrnmpt.1 | |
ralrnmpt.2 |
Ref | Expression |
---|---|
ralrnmpt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrnmpt.1 | . . . . 5 | |
2 | 1 | fnmpt 5249 | . . . 4 |
3 | dfsbcq 2911 | . . . . 5 | |
4 | 3 | ralrn 5558 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 |
6 | nfv 1508 | . . . . 5 | |
7 | nfsbc1v 2927 | . . . . 5 | |
8 | sbceq1a 2918 | . . . . 5 | |
9 | 6, 7, 8 | cbvral 2650 | . . . 4 |
10 | 9 | bicomi 131 | . . 3 |
11 | nfmpt1 4021 | . . . . . . 7 | |
12 | 1, 11 | nfcxfr 2278 | . . . . . 6 |
13 | nfcv 2281 | . . . . . 6 | |
14 | 12, 13 | nffv 5431 | . . . . 5 |
15 | nfv 1508 | . . . . 5 | |
16 | 14, 15 | nfsbc 2929 | . . . 4 |
17 | nfv 1508 | . . . 4 | |
18 | fveq2 5421 | . . . . 5 | |
19 | dfsbcq 2911 | . . . . 5 | |
20 | 18, 19 | syl 14 | . . . 4 |
21 | 16, 17, 20 | cbvral 2650 | . . 3 |
22 | 5, 10, 21 | 3bitr3g 221 | . 2 |
23 | 1 | fvmpt2 5504 | . . . . . 6 |
24 | dfsbcq 2911 | . . . . . 6 | |
25 | 23, 24 | syl 14 | . . . . 5 |
26 | ralrnmpt.2 | . . . . . . 7 | |
27 | 26 | sbcieg 2941 | . . . . . 6 |
28 | 27 | adantl 275 | . . . . 5 |
29 | 25, 28 | bitrd 187 | . . . 4 |
30 | 29 | ralimiaa 2494 | . . 3 |
31 | ralbi 2564 | . . 3 | |
32 | 30, 31 | syl 14 | . 2 |
33 | 22, 32 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wsbc 2909 cmpt 3989 crn 4540 wfn 5118 cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: (None) |
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