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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 5727 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 2081 | . 2 | |
2 | imassrn 4699 | . . . . 5 | |
3 | f1opw2.1 | . . . . . . 7 | |
4 | f1ofo 5153 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | forn 5129 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 |
8 | 2, 7 | syl5sseq 3047 | . . . 4 |
9 | f1opw2.3 | . . . . 5 | |
10 | elpwg 3390 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 8, 11 | mpbird 165 | . . 3 |
13 | 12 | adantr 270 | . 2 |
14 | imassrn 4699 | . . . . 5 | |
15 | dfdm4 4545 | . . . . . 6 | |
16 | f1odm 5150 | . . . . . . 7 | |
17 | 3, 16 | syl 14 | . . . . . 6 |
18 | 15, 17 | syl5eqr 2127 | . . . . 5 |
19 | 14, 18 | syl5sseq 3047 | . . . 4 |
20 | f1opw2.2 | . . . . 5 | |
21 | elpwg 3390 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | 19, 22 | mpbird 165 | . . 3 |
24 | 23 | adantr 270 | . 2 |
25 | elpwi 3391 | . . . . . . 7 | |
26 | 25 | adantl 271 | . . . . . 6 |
27 | foimacnv 5164 | . . . . . 6 | |
28 | 5, 26, 27 | syl2an 283 | . . . . 5 |
29 | 28 | eqcomd 2086 | . . . 4 |
30 | imaeq2 4684 | . . . . 5 | |
31 | 30 | eqeq2d 2092 | . . . 4 |
32 | 29, 31 | syl5ibrcom 155 | . . 3 |
33 | f1of1 5145 | . . . . . . 7 | |
34 | 3, 33 | syl 14 | . . . . . 6 |
35 | elpwi 3391 | . . . . . . 7 | |
36 | 35 | adantr 270 | . . . . . 6 |
37 | f1imacnv 5163 | . . . . . 6 | |
38 | 34, 36, 37 | syl2an 283 | . . . . 5 |
39 | 38 | eqcomd 2086 | . . . 4 |
40 | imaeq2 4684 | . . . . 5 | |
41 | 40 | eqeq2d 2092 | . . . 4 |
42 | 39, 41 | syl5ibrcom 155 | . . 3 |
43 | 32, 42 | impbid 127 | . 2 |
44 | 1, 13, 24, 43 | f1o2d 5725 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cvv 2601 wss 2973 cpw 3382 cmpt 3839 ccnv 4362 cdm 4363 crn 4364 cima 4366 wf1 4919 wfo 4920 wf1o 4921 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 |
This theorem is referenced by: f1opw 5727 |
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