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Mirrors > Home > ILE Home > Th. List > f1imaeng | Unicode version |
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
f1imaeng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ores 5442 | . . 3 | |
2 | f1oeng 6715 | . . . 4 | |
3 | 2 | ancoms 266 | . . 3 |
4 | 1, 3 | stoic3 1418 | . 2 |
5 | 4 | ensymd 6741 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 967 wcel 2135 wss 3112 class class class wbr 3977 cres 4601 cima 4602 wf1 5180 wf1o 5182 cen 6696 |
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-coll 4092 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 |
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-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-er 6493 df-en 6699 |
This theorem is referenced by: f1imaen 6752 isinfinf 6855 f1finf1o 6904 phimullem 12143 |
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