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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 5945 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 2117 | . 2 | |
2 | imassrn 4862 | . . . . 5 | |
3 | f1opw2.1 | . . . . . . 7 | |
4 | f1ofo 5342 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | forn 5318 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 |
8 | 2, 7 | sseqtrid 3117 | . . . 4 |
9 | f1opw2.3 | . . . . 5 | |
10 | elpwg 3488 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 8, 11 | mpbird 166 | . . 3 |
13 | 12 | adantr 274 | . 2 |
14 | imassrn 4862 | . . . . 5 | |
15 | dfdm4 4701 | . . . . . 6 | |
16 | f1odm 5339 | . . . . . . 7 | |
17 | 3, 16 | syl 14 | . . . . . 6 |
18 | 15, 17 | syl5eqr 2164 | . . . . 5 |
19 | 14, 18 | sseqtrid 3117 | . . . 4 |
20 | f1opw2.2 | . . . . 5 | |
21 | elpwg 3488 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | 19, 22 | mpbird 166 | . . 3 |
24 | 23 | adantr 274 | . 2 |
25 | elpwi 3489 | . . . . . . 7 | |
26 | 25 | adantl 275 | . . . . . 6 |
27 | foimacnv 5353 | . . . . . 6 | |
28 | 5, 26, 27 | syl2an 287 | . . . . 5 |
29 | 28 | eqcomd 2123 | . . . 4 |
30 | imaeq2 4847 | . . . . 5 | |
31 | 30 | eqeq2d 2129 | . . . 4 |
32 | 29, 31 | syl5ibrcom 156 | . . 3 |
33 | f1of1 5334 | . . . . . . 7 | |
34 | 3, 33 | syl 14 | . . . . . 6 |
35 | elpwi 3489 | . . . . . . 7 | |
36 | 35 | adantr 274 | . . . . . 6 |
37 | f1imacnv 5352 | . . . . . 6 | |
38 | 34, 36, 37 | syl2an 287 | . . . . 5 |
39 | 38 | eqcomd 2123 | . . . 4 |
40 | imaeq2 4847 | . . . . 5 | |
41 | 40 | eqeq2d 2129 | . . . 4 |
42 | 39, 41 | syl5ibrcom 156 | . . 3 |
43 | 32, 42 | impbid 128 | . 2 |
44 | 1, 13, 24, 43 | f1o2d 5943 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cvv 2660 wss 3041 cpw 3480 cmpt 3959 ccnv 4508 cdm 4509 crn 4510 cima 4512 wf1 5090 wfo 5091 wf1o 5092 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 |
This theorem is referenced by: f1opw 5945 |
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