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Mirrors > Home > ILE Home > Th. List > cofunexg | 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 4762 |
. . 3
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2 | relssdmrn 4784 |
. . 3
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3 | 1, 2 | ax-mp 7 |
. 2
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4 | dmcoss 4544 |
. . . . 5
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5 | dmexg 4539 |
. . . . 5
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6 | ssexg 3887 |
. . . . 5
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7 | 4, 5, 6 | sylancr 393 |
. . . 4
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8 | 7 | adantl 262 |
. . 3
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9 | rnco 4770 |
. . . 4
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10 | rnexg 4540 |
. . . . . 6
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11 | resfunexg 5325 |
. . . . . 6
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12 | 10, 11 | sylan2 270 |
. . . . 5
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13 | rnexg 4540 |
. . . . 5
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14 | 12, 13 | syl 14 |
. . . 4
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15 | 9, 14 | syl5eqel 2121 |
. . 3
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16 | xpexg 4395 |
. . 3
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17 | 8, 15, 16 | syl2anc 391 |
. 2
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18 | ssexg 3887 |
. 2
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19 | 3, 17, 18 | sylancr 393 |
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 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 |
This theorem is referenced by: cofunex2g 5681 |
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