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Mirrors > Home > MPE Home > Th. List > Mathboxes > indexa | Structured version Visualization version Unicode version |
Description: If for every element of an indexing set there exists a corresponding element of another set , then there exists a subset of consisting only of those elements which are indexed by . Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
indexa |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabexg 4812 | . 2 | |
2 | ssrab2 3687 | . . . 4 | |
3 | 2 | a1i 11 | . . 3 |
4 | nfv 1843 | . . . . 5 | |
5 | nfre1 3005 | . . . . 5 | |
6 | sbceq2a 3447 | . . . . . . . . . . . . . . 15 | |
7 | 6 | rspcev 3309 | . . . . . . . . . . . . . 14 |
8 | 7 | ancoms 469 | . . . . . . . . . . . . 13 |
9 | 8 | anim2i 593 | . . . . . . . . . . . 12 |
10 | 9 | ancoms 469 | . . . . . . . . . . 11 |
11 | 10 | anasss 679 | . . . . . . . . . 10 |
12 | 11 | ancoms 469 | . . . . . . . . 9 |
13 | sbceq2a 3447 | . . . . . . . . . . . 12 | |
14 | 13 | sbcbidv 3490 | . . . . . . . . . . 11 |
15 | 14 | rexbidv 3052 | . . . . . . . . . 10 |
16 | 15 | elrab 3363 | . . . . . . . . 9 |
17 | 12, 16 | sylibr 224 | . . . . . . . 8 |
18 | sbceq2a 3447 | . . . . . . . . 9 | |
19 | 18 | rspcev 3309 | . . . . . . . 8 |
20 | 17, 19 | sylancom 701 | . . . . . . 7 |
21 | nfcv 2764 | . . . . . . . 8 | |
22 | nfcv 2764 | . . . . . . . . . 10 | |
23 | nfcv 2764 | . . . . . . . . . . 11 | |
24 | nfsbc1v 3455 | . . . . . . . . . . 11 | |
25 | 23, 24 | nfsbc 3457 | . . . . . . . . . 10 |
26 | 22, 25 | nfrex 3007 | . . . . . . . . 9 |
27 | nfcv 2764 | . . . . . . . . 9 | |
28 | 26, 27 | nfrab 3123 | . . . . . . . 8 |
29 | nfsbc1v 3455 | . . . . . . . 8 | |
30 | nfv 1843 | . . . . . . . 8 | |
31 | 21, 28, 29, 30, 18 | cbvrexf 3166 | . . . . . . 7 |
32 | 20, 31 | sylib 208 | . . . . . 6 |
33 | 32 | exp31 630 | . . . . 5 |
34 | 4, 5, 33 | rexlimd 3026 | . . . 4 |
35 | 34 | ralimia 2950 | . . 3 |
36 | nfsbc1v 3455 | . . . . . . . . 9 | |
37 | nfv 1843 | . . . . . . . . 9 | |
38 | 36, 37, 6 | cbvrex 3168 | . . . . . . . 8 |
39 | 15, 38 | syl6bb 276 | . . . . . . 7 |
40 | 39 | elrab 3363 | . . . . . 6 |
41 | 40 | simprbi 480 | . . . . 5 |
42 | 41 | rgen 2922 | . . . 4 |
43 | 42 | a1i 11 | . . 3 |
44 | 3, 35, 43 | 3jca 1242 | . 2 |
45 | sseq1 3626 | . . . . 5 | |
46 | nfcv 2764 | . . . . . . . . 9 | |
47 | nfsbc1v 3455 | . . . . . . . . 9 | |
48 | 46, 47 | nfrex 3007 | . . . . . . . 8 |
49 | nfcv 2764 | . . . . . . . 8 | |
50 | 48, 49 | nfrab 3123 | . . . . . . 7 |
51 | 50 | nfeq2 2780 | . . . . . 6 |
52 | nfcv 2764 | . . . . . . 7 | |
53 | 52, 28 | rexeqf 3135 | . . . . . 6 |
54 | 51, 53 | ralbid 2983 | . . . . 5 |
55 | 52, 28 | raleqf 3134 | . . . . 5 |
56 | 45, 54, 55 | 3anbi123d 1399 | . . . 4 |
57 | 56 | spcegv 3294 | . . 3 |
58 | 57 | imp 445 | . 2 |
59 | 1, 44, 58 | syl2an 494 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wex 1704 wcel 1990 wral 2912 wrex 2913 crab 2916 cvv 3200 wsbc 3435 wss 3574 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
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-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-in 3581 df-ss 3588 |
This theorem is referenced by: (None) |
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