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Mirrors > Home > ILE Home > Th. List > f1opw2 | Unicode version |
Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5707 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
f1opw2.1 | |
f1opw2.2 | |
f1opw2.3 |
Ref | Expression |
---|---|
f1opw2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2040 | . 2 | |
2 | imassrn 4679 | . . . . 5 | |
3 | f1opw2.1 | . . . . . . 7 | |
4 | f1ofo 5133 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | forn 5109 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 |
8 | 2, 7 | syl5sseq 2993 | . . . 4 |
9 | f1opw2.3 | . . . . 5 | |
10 | elpwg 3367 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 8, 11 | mpbird 156 | . . 3 |
13 | 12 | adantr 261 | . 2 |
14 | imassrn 4679 | . . . . 5 | |
15 | dfdm4 4527 | . . . . . 6 | |
16 | f1odm 5130 | . . . . . . 7 | |
17 | 3, 16 | syl 14 | . . . . . 6 |
18 | 15, 17 | syl5eqr 2086 | . . . . 5 |
19 | 14, 18 | syl5sseq 2993 | . . . 4 |
20 | f1opw2.2 | . . . . 5 | |
21 | elpwg 3367 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | 19, 22 | mpbird 156 | . . 3 |
24 | 23 | adantr 261 | . 2 |
25 | elpwi 3368 | . . . . . . 7 | |
26 | 25 | adantl 262 | . . . . . 6 |
27 | foimacnv 5144 | . . . . . 6 | |
28 | 5, 26, 27 | syl2an 273 | . . . . 5 |
29 | 28 | eqcomd 2045 | . . . 4 |
30 | imaeq2 4664 | . . . . 5 | |
31 | 30 | eqeq2d 2051 | . . . 4 |
32 | 29, 31 | syl5ibrcom 146 | . . 3 |
33 | f1of1 5125 | . . . . . . 7 | |
34 | 3, 33 | syl 14 | . . . . . 6 |
35 | elpwi 3368 | . . . . . . 7 | |
36 | 35 | adantr 261 | . . . . . 6 |
37 | f1imacnv 5143 | . . . . . 6 | |
38 | 34, 36, 37 | syl2an 273 | . . . . 5 |
39 | 38 | eqcomd 2045 | . . . 4 |
40 | imaeq2 4664 | . . . . 5 | |
41 | 40 | eqeq2d 2051 | . . . 4 |
42 | 39, 41 | syl5ibrcom 146 | . . 3 |
43 | 32, 42 | impbid 120 | . 2 |
44 | 1, 13, 24, 43 | f1o2d 5705 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 cvv 2557 wss 2917 cpw 3359 cmpt 3818 ccnv 4344 cdm 4345 crn 4346 cima 4348 wf1 4899 wfo 4900 wf1o 4901 |
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-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 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-res 4357 df-ima 4358 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1opw 5707 |
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