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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 11671 |
. . 3
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3 | setsn0fun.i |
. . . . 5
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4 | setsn0fun.e |
. . . . 5
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5 | structex 11670 |
. . . . . . 7
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6 | setsfun0 11694 |
. . . . . . 7
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7 | 5, 6 | sylanl1 395 |
. . . . . 6
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8 | 7 | expcom 115 |
. . . . 5
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9 | 3, 4, 8 | syl2anc 404 |
. . . 4
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10 | 9 | com12 30 |
. . 3
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11 | 2, 10 | mpdan 413 |
. 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-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-res 4479 df-iota 5014 df-fun 5051 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-struct 11660 df-sets 11665 |
This theorem is referenced by: (None) |
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