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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 5025 | . . . 4 |
3 | dfsbcq 2766 | . . . . 5 | |
4 | 3 | ralrn 5305 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 |
6 | nfv 1421 | . . . . 5 | |
7 | nfsbc1v 2782 | . . . . 5 | |
8 | sbceq1a 2773 | . . . . 5 | |
9 | 6, 7, 8 | cbvral 2529 | . . . 4 |
10 | 9 | bicomi 123 | . . 3 |
11 | nfmpt1 3850 | . . . . . . 7 | |
12 | 1, 11 | nfcxfr 2175 | . . . . . 6 |
13 | nfcv 2178 | . . . . . 6 | |
14 | 12, 13 | nffv 5185 | . . . . 5 |
15 | nfv 1421 | . . . . 5 | |
16 | 14, 15 | nfsbc 2784 | . . . 4 |
17 | nfv 1421 | . . . 4 | |
18 | fveq2 5178 | . . . . 5 | |
19 | dfsbcq 2766 | . . . . 5 | |
20 | 18, 19 | syl 14 | . . . 4 |
21 | 16, 17, 20 | cbvral 2529 | . . 3 |
22 | 5, 10, 21 | 3bitr3g 211 | . 2 |
23 | 1 | fvmpt2 5254 | . . . . . 6 |
24 | dfsbcq 2766 | . . . . . 6 | |
25 | 23, 24 | syl 14 | . . . . 5 |
26 | ralrnmpt.2 | . . . . . . 7 | |
27 | 26 | sbcieg 2795 | . . . . . 6 |
28 | 27 | adantl 262 | . . . . 5 |
29 | 25, 28 | bitrd 177 | . . . 4 |
30 | 29 | ralimiaa 2383 | . . 3 |
31 | ralbi 2445 | . . 3 | |
32 | 30, 31 | syl 14 | . 2 |
33 | 22, 32 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 wsbc 2764 cmpt 3818 crn 4346 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: (None) |
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