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Mirrors > Home > ILE Home > Th. List > rexrnmpt | 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 |
---|---|
rexrnmpt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrnmpt.1 | . . . . 5 | |
2 | 1 | fnmpt 5244 | . . . 4 |
3 | dfsbcq 2906 | . . . . 5 | |
4 | 3 | rexrn 5550 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 |
6 | nfv 1508 | . . . . 5 | |
7 | nfsbc1v 2922 | . . . . 5 | |
8 | sbceq1a 2913 | . . . . 5 | |
9 | 6, 7, 8 | cbvrex 2649 | . . . 4 |
10 | 9 | bicomi 131 | . . 3 |
11 | nfmpt1 4016 | . . . . . . 7 | |
12 | 1, 11 | nfcxfr 2276 | . . . . . 6 |
13 | nfcv 2279 | . . . . . 6 | |
14 | 12, 13 | nffv 5424 | . . . . 5 |
15 | nfv 1508 | . . . . 5 | |
16 | 14, 15 | nfsbc 2924 | . . . 4 |
17 | nfv 1508 | . . . 4 | |
18 | fveq2 5414 | . . . . 5 | |
19 | 18 | sbceq1d 2909 | . . . 4 |
20 | 16, 17, 19 | cbvrex 2649 | . . 3 |
21 | 5, 10, 20 | 3bitr3g 221 | . 2 |
22 | 1 | fvmpt2 5497 | . . . . . 6 |
23 | 22 | sbceq1d 2909 | . . . . 5 |
24 | ralrnmpt.2 | . . . . . . 7 | |
25 | 24 | sbcieg 2936 | . . . . . 6 |
26 | 25 | adantl 275 | . . . . 5 |
27 | 23, 26 | bitrd 187 | . . . 4 |
28 | 27 | ralimiaa 2492 | . . 3 |
29 | pm5.32 448 | . . . . . 6 | |
30 | 29 | albii 1446 | . . . . 5 |
31 | exbi 1583 | . . . . 5 | |
32 | 30, 31 | sylbi 120 | . . . 4 |
33 | df-ral 2419 | . . . 4 | |
34 | df-rex 2420 | . . . . 5 | |
35 | df-rex 2420 | . . . . 5 | |
36 | 34, 35 | bibi12i 228 | . . . 4 |
37 | 32, 33, 36 | 3imtr4i 200 | . . 3 |
38 | 28, 37 | syl 14 | . 2 |
39 | 21, 38 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wex 1468 wcel 1480 wral 2414 wrex 2415 wsbc 2904 cmpt 3984 crn 4535 wfn 5113 cfv 5118 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: txbas 12416 |
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