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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 6794 does not need ax-setind 4537.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
f1imaen2g |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 531 |
. . 3
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2 | simplr 528 |
. . . 4
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3 | f1f 5422 |
. . . . . 6
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4 | imassrn 4982 |
. . . . . . 7
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5 | frn 5375 |
. . . . . . 7
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6 | 4, 5 | sstrid 3167 |
. . . . . 6
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7 | 3, 6 | syl 14 |
. . . . 5
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8 | 7 | ad2antrr 488 |
. . . 4
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9 | 2, 8 | ssexd 4144 |
. . 3
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10 | f1ores 5477 |
. . . 4
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11 | 10 | ad2ant2r 509 |
. . 3
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12 | f1oen2g 6755 |
. . 3
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13 | 1, 9, 11, 12 | syl3anc 1238 |
. 2
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14 | 13 | ensymd 6783 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-er 6535 df-en 6741 |
This theorem is referenced by: ssenen 6851 phplem4 6855 phplem4dom 6862 phplem4on 6867 fiintim 6928 |
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