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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 6341 does not need ax-setind 4288.) (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 499 |
. . 3
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2 | simplr 497 |
. . . 4
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3 | f1f 5123 |
. . . . . 6
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4 | imassrn 4709 |
. . . . . . 7
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5 | frn 5083 |
. . . . . . 7
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6 | 4, 5 | syl5ss 3011 |
. . . . . 6
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7 | 3, 6 | syl 14 |
. . . . 5
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8 | 7 | ad2antrr 472 |
. . . 4
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9 | 2, 8 | ssexd 3926 |
. . 3
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10 | f1ores 5172 |
. . . 4
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11 | 10 | ad2ant2r 493 |
. . 3
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12 | f1oen2g 6302 |
. . 3
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13 | 1, 9, 11, 12 | syl3anc 1170 |
. 2
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14 | 13 | ensymd 6330 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-er 6172 df-en 6288 |
This theorem is referenced by: phplem4 6390 phplem4dom 6397 phplem4on 6402 |
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