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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 6751 does not need ax-setind 4508.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
f1imaen2g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 522 | . . 3 | |
2 | simplr 520 | . . . 4 | |
3 | f1f 5387 | . . . . . 6 | |
4 | imassrn 4951 | . . . . . . 7 | |
5 | frn 5340 | . . . . . . 7 | |
6 | 4, 5 | sstrid 3148 | . . . . . 6 |
7 | 3, 6 | syl 14 | . . . . 5 |
8 | 7 | ad2antrr 480 | . . . 4 |
9 | 2, 8 | ssexd 4116 | . . 3 |
10 | f1ores 5441 | . . . 4 | |
11 | 10 | ad2ant2r 501 | . . 3 |
12 | f1oen2g 6712 | . . 3 | |
13 | 1, 9, 11, 12 | syl3anc 1227 | . 2 |
14 | 13 | ensymd 6740 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 cvv 2721 wss 3111 class class class wbr 3976 crn 4599 cres 4600 cima 4601 wf 5178 wf1 5179 wf1o 5181 cen 6695 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-er 6492 df-en 6698 |
This theorem is referenced by: ssenen 6808 phplem4 6812 phplem4dom 6819 phplem4on 6824 fiintim 6885 |
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