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Mirrors > Home > ILE Home > Th. List > f1imaen2g | Unicode version |
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6688 does not need ax-setind 4452.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
f1imaen2g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 521 | . . 3 | |
2 | simplr 519 | . . . 4 | |
3 | f1f 5328 | . . . . . 6 | |
4 | imassrn 4892 | . . . . . . 7 | |
5 | frn 5281 | . . . . . . 7 | |
6 | 4, 5 | sstrid 3108 | . . . . . 6 |
7 | 3, 6 | syl 14 | . . . . 5 |
8 | 7 | ad2antrr 479 | . . . 4 |
9 | 2, 8 | ssexd 4068 | . . 3 |
10 | f1ores 5382 | . . . 4 | |
11 | 10 | ad2ant2r 500 | . . 3 |
12 | f1oen2g 6649 | . . 3 | |
13 | 1, 9, 11, 12 | syl3anc 1216 | . 2 |
14 | 13 | ensymd 6677 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 cvv 2686 wss 3071 class class class wbr 3929 crn 4540 cres 4541 cima 4542 wf 5119 wf1 5120 wf1o 5122 cen 6632 |
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-13 1491 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 ax-un 4355 |
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-uni 3737 df-br 3930 df-opab 3990 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 df-er 6429 df-en 6635 |
This theorem is referenced by: ssenen 6745 phplem4 6749 phplem4dom 6756 phplem4on 6761 fiintim 6817 |
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