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Mirrors > Home > ILE Home > Th. List > setsn0fun | Unicode version |
Description: The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
Ref | Expression |
---|---|
setsn0fun.s |
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setsn0fun.i |
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setsn0fun.e |
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Ref | Expression |
---|---|
setsn0fun |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsn0fun.s |
. 2
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2 | structn0fun 12458 |
. . 3
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3 | setsn0fun.i |
. . . . 5
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4 | setsn0fun.e |
. . . . 5
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5 | structex 12457 |
. . . . . . 7
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6 | setsfun0 12481 |
. . . . . . 7
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7 | 5, 6 | sylanl1 402 |
. . . . . 6
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8 | 7 | expcom 116 |
. . . . 5
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9 | 3, 4, 8 | syl2anc 411 |
. . . 4
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10 | 9 | com12 30 |
. . 3
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11 | 2, 10 | mpdan 421 |
. 2
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12 | 1, 11 | mpcom 36 |
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-in1 614 ax-in2 615 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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-res 4635 df-iota 5174 df-fun 5214 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-struct 12447 df-sets 12452 |
This theorem is referenced by: (None) |
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