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Mirrors > Home > MPE Home > Th. List > resixp | Structured version 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 5256 |
. . 3
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2 | 1 | adantl 466 |
. 2
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3 | simpr 461 |
. . . . 5
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4 | elixp2 7376 |
. . . . 5
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5 | 3, 4 | sylib 196 |
. . . 4
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6 | 5 | simp2d 1001 |
. . 3
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7 | simpl 457 |
. . 3
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8 | fnssres 5631 |
. . 3
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9 | 6, 7, 8 | syl2anc 661 |
. 2
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10 | 5 | simp3d 1002 |
. . . 4
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11 | ssralv 3523 |
. . . 4
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12 | 7, 10, 11 | sylc 60 |
. . 3
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13 | fvres 5812 |
. . . . 5
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14 | 13 | eleq1d 2523 |
. . . 4
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15 | 14 | ralbiia 2837 |
. . 3
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16 | 12, 15 | sylibr 212 |
. 2
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17 | elixp2 7376 |
. 2
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18 | 2, 9, 16, 17 | syl3anbrc 1172 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4520 ax-nul 4528 ax-pr 4638 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-ral 2803 df-rex 2804 df-rab 2807 df-v 3078 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-nul 3745 df-if 3899 df-sn 3985 df-pr 3987 df-op 3991 df-uni 4199 df-br 4400 df-opab 4458 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-res 4959 df-iota 5488 df-fun 5527 df-fn 5528 df-fv 5533 df-ixp 7373 |
This theorem is referenced by: resixpfo 7410 ixpfi2 7719 ptrescn 19343 ptuncnv 19511 ptcmplem2 19756 |
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