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Mirrors > Home > ILE Home > Th. List > sefvex | Unicode version |
Description: If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
sefvex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2692 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
2 | 1 | a1i 9 |
. . . . . . 7
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3 | simp3 984 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | simp2 983 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | brcnvg 4728 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 1, 4, 5 | sylancr 411 |
. . . . . . . 8
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7 | 3, 6 | mpbird 166 |
. . . . . . 7
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8 | breq1 3940 |
. . . . . . . 8
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9 | 8 | elrab 2844 |
. . . . . . 7
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10 | 2, 7, 9 | sylanbrc 414 |
. . . . . 6
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11 | elssuni 3772 |
. . . . . 6
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12 | 10, 11 | syl 14 |
. . . . 5
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13 | 12 | 3expia 1184 |
. . . 4
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14 | 13 | alrimiv 1847 |
. . 3
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15 | fvss 5443 |
. . 3
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16 | 14, 15 | syl 14 |
. 2
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17 | seex 4265 |
. . 3
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18 | uniexg 4369 |
. . 3
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19 | 17, 18 | syl 14 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | ssexg 4075 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 16, 19, 20 | syl2anc 409 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-se 4263 df-cnv 4555 df-iota 5096 df-fv 5139 |
This theorem is referenced by: (None) |
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