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Mirrors > Home > MPE Home > Th. List > fresaunres1 | Structured version Unicode version |
Description: From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.) |
Ref | Expression |
---|---|
fresaunres1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3601 |
. . 3
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2 | 1 | reseq1i 5207 |
. 2
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3 | incom 3644 |
. . . . . 6
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4 | 3 | reseq2i 5208 |
. . . . 5
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5 | 3 | reseq2i 5208 |
. . . . 5
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6 | 4, 5 | eqeq12i 2471 |
. . . 4
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7 | eqcom 2460 |
. . . 4
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8 | 6, 7 | bitri 249 |
. . 3
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9 | fresaunres2 5684 |
. . . 4
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10 | 9 | 3com12 1192 |
. . 3
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11 | 8, 10 | syl3an3b 1257 |
. 2
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12 | 2, 11 | syl5eq 2504 |
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 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pr 4632 |
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 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3073 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-sn 3979 df-pr 3981 df-op 3985 df-br 4394 df-opab 4452 df-xp 4947 df-rel 4948 df-dm 4951 df-res 4953 df-fun 5521 df-fn 5522 df-f 5523 |
This theorem is referenced by: mapunen 7583 hashf1lem1 12319 ptuncnv 19505 resf1o 26174 cvmliftlem10 27320 |
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