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Mirrors > Home > ILE Home > Th. List > resixp | Unicode version |
Description: Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.) |
Ref | Expression |
---|---|
resixp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resexg 4785 |
. . 3
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2 | 1 | adantl 272 |
. 2
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3 | simpr 109 |
. . . . 5
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4 | elixp2 6499 |
. . . . 5
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5 | 3, 4 | sylib 121 |
. . . 4
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6 | 5 | simp2d 959 |
. . 3
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7 | simpl 108 |
. . 3
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8 | fnssres 5161 |
. . 3
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9 | 6, 7, 8 | syl2anc 404 |
. 2
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10 | 5 | simp3d 960 |
. . . 4
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11 | ssralv 3100 |
. . . 4
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12 | 7, 10, 11 | sylc 62 |
. . 3
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13 | fvres 5364 |
. . . . 5
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14 | 13 | eleq1d 2163 |
. . . 4
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15 | 14 | ralbiia 2403 |
. . 3
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16 | 12, 15 | sylibr 133 |
. 2
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17 | elixp2 6499 |
. 2
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18 | 2, 9, 16, 17 | syl3anbrc 1130 |
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-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-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 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 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-fn 5052 df-fv 5057 df-ixp 6496 |
This theorem is referenced by: (None) |
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