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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 5977 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 2139 | . 2 | |
2 | imassrn 4892 | . . . . 5 | |
3 | f1opw2.1 | . . . . . . 7 | |
4 | f1ofo 5374 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | forn 5348 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 |
8 | 2, 7 | sseqtrid 3147 | . . . 4 |
9 | f1opw2.3 | . . . . 5 | |
10 | elpwg 3518 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 8, 11 | mpbird 166 | . . 3 |
13 | 12 | adantr 274 | . 2 |
14 | imassrn 4892 | . . . . 5 | |
15 | dfdm4 4731 | . . . . . 6 | |
16 | f1odm 5371 | . . . . . . 7 | |
17 | 3, 16 | syl 14 | . . . . . 6 |
18 | 15, 17 | syl5eqr 2186 | . . . . 5 |
19 | 14, 18 | sseqtrid 3147 | . . . 4 |
20 | f1opw2.2 | . . . . 5 | |
21 | elpwg 3518 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | 19, 22 | mpbird 166 | . . 3 |
24 | 23 | adantr 274 | . 2 |
25 | elpwi 3519 | . . . . . . 7 | |
26 | 25 | adantl 275 | . . . . . 6 |
27 | foimacnv 5385 | . . . . . 6 | |
28 | 5, 26, 27 | syl2an 287 | . . . . 5 |
29 | 28 | eqcomd 2145 | . . . 4 |
30 | imaeq2 4877 | . . . . 5 | |
31 | 30 | eqeq2d 2151 | . . . 4 |
32 | 29, 31 | syl5ibrcom 156 | . . 3 |
33 | f1of1 5366 | . . . . . . 7 | |
34 | 3, 33 | syl 14 | . . . . . 6 |
35 | elpwi 3519 | . . . . . . 7 | |
36 | 35 | adantr 274 | . . . . . 6 |
37 | f1imacnv 5384 | . . . . . 6 | |
38 | 34, 36, 37 | syl2an 287 | . . . . 5 |
39 | 38 | eqcomd 2145 | . . . 4 |
40 | imaeq2 4877 | . . . . 5 | |
41 | 40 | eqeq2d 2151 | . . . 4 |
42 | 39, 41 | syl5ibrcom 156 | . . 3 |
43 | 32, 42 | impbid 128 | . 2 |
44 | 1, 13, 24, 43 | f1o2d 5975 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cvv 2686 wss 3071 cpw 3510 cmpt 3989 ccnv 4538 cdm 4539 crn 4540 cima 4542 wf1 5120 wfo 5121 wf1o 5122 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: f1opw 5977 |
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