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Mirrors > Home > MPE Home > Th. List > cofunexg | Structured version Visualization version Unicode version |
Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.) |
Ref | Expression |
---|---|
cofunexg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5333 |
. . 3
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2 | relssdmrn 5356 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
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4 | dmcoss 5094 |
. . . . 5
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5 | dmexg 6724 |
. . . . 5
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6 | ssexg 4549 |
. . . . 5
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7 | 4, 5, 6 | sylancr 669 |
. . . 4
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8 | 7 | adantl 468 |
. . 3
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9 | rnco 5341 |
. . . 4
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10 | rnexg 6725 |
. . . . . 6
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11 | resfunexg 6130 |
. . . . . 6
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12 | 10, 11 | sylan2 477 |
. . . . 5
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13 | rnexg 6725 |
. . . . 5
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14 | 12, 13 | syl 17 |
. . . 4
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15 | 9, 14 | syl5eqel 2533 |
. . 3
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16 | xpexg 6593 |
. . 3
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17 | 8, 15, 16 | syl2anc 667 |
. 2
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18 | ssexg 4549 |
. 2
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19 | 3, 17, 18 | sylancr 669 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 |
This theorem is referenced by: cofunex2g 6758 fin1a2lem7 8836 revco 12931 ccatco 12932 lswco 12935 isofval 15662 bcthlem4 22295 sseqval 29221 sinccvglem 30316 pfxco 38979 |
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