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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 4968 |
. . 3
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2 | 1 | adantl 277 |
. 2
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3 | simpr 110 |
. . . . 5
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4 | elixp2 6732 |
. . . . 5
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5 | 3, 4 | sylib 122 |
. . . 4
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6 | 5 | simp2d 1012 |
. . 3
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7 | simpl 109 |
. . 3
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8 | fnssres 5351 |
. . 3
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9 | 6, 7, 8 | syl2anc 411 |
. 2
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10 | 5 | simp3d 1013 |
. . . 4
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11 | ssralv 3234 |
. . . 4
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12 | 7, 10, 11 | sylc 62 |
. . 3
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13 | fvres 5561 |
. . . . 5
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14 | 13 | eleq1d 2258 |
. . . 4
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15 | 14 | ralbiia 2504 |
. . 3
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16 | 12, 15 | sylibr 134 |
. 2
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17 | elixp2 6732 |
. 2
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18 | 2, 9, 16, 17 | syl3anbrc 1183 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-res 4659 df-iota 5199 df-fun 5240 df-fn 5241 df-fv 5246 df-ixp 6729 |
This theorem is referenced by: (None) |
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