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Mirrors > Home > MPE Home > Th. List > ralrnmpt | Structured version Visualization version 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 6020 | . . . 4 |
3 | dfsbcq 3437 | . . . . 5 | |
4 | 3 | ralrn 6362 | . . . 4 |
5 | 2, 4 | syl 17 | . . 3 |
6 | nfv 1843 | . . . . 5 | |
7 | nfsbc1v 3455 | . . . . 5 | |
8 | sbceq1a 3446 | . . . . 5 | |
9 | 6, 7, 8 | cbvral 3167 | . . . 4 |
10 | 9 | bicomi 214 | . . 3 |
11 | nfmpt1 4747 | . . . . . . 7 | |
12 | 1, 11 | nfcxfr 2762 | . . . . . 6 |
13 | nfcv 2764 | . . . . . 6 | |
14 | 12, 13 | nffv 6198 | . . . . 5 |
15 | nfv 1843 | . . . . 5 | |
16 | 14, 15 | nfsbc 3457 | . . . 4 |
17 | nfv 1843 | . . . 4 | |
18 | fveq2 6191 | . . . . 5 | |
19 | 18 | sbceq1d 3440 | . . . 4 |
20 | 16, 17, 19 | cbvral 3167 | . . 3 |
21 | 5, 10, 20 | 3bitr3g 302 | . 2 |
22 | 1 | fvmpt2 6291 | . . . . . 6 |
23 | 22 | sbceq1d 3440 | . . . . 5 |
24 | ralrnmpt.2 | . . . . . . 7 | |
25 | 24 | sbcieg 3468 | . . . . . 6 |
26 | 25 | adantl 482 | . . . . 5 |
27 | 23, 26 | bitrd 268 | . . . 4 |
28 | 27 | ralimiaa 2951 | . . 3 |
29 | ralbi 3068 | . . 3 | |
30 | 28, 29 | syl 17 | . 2 |
31 | 21, 30 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wsbc 3435 cmpt 4729 crn 5115 wfn 5883 cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: rexrnmpt 6369 ac6num 9301 gsumwspan 17383 dfod2 17981 ordtbaslem 20992 ordtrest2lem 21007 cncmp 21195 comppfsc 21335 ptpjopn 21415 ordthmeolem 21604 tsmsfbas 21931 tsmsf1o 21948 prdsxmetlem 22173 prdsbl 22296 metdsf 22651 metdsge 22652 minveclem1 23195 minveclem3b 23199 minveclem6 23205 mbflimsup 23433 xrlimcnp 24695 minvecolem1 27730 minvecolem5 27737 minvecolem6 27738 ordtrest2NEWlem 29968 cvmsss2 31256 fin2so 33396 prdsbnd 33592 rrnequiv 33634 ralrnmpt3 39474 |
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