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Mirrors > Home > MPE Home > Th. List > foco | Structured version Visualization version Unicode version |
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
foco |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 5810 |
. . 3
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2 | dffo2 5810 |
. . 3
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3 | fco 5751 |
. . . . 5
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4 | 3 | ad2ant2r 761 |
. . . 4
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5 | fdm 5745 |
. . . . . . . 8
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6 | eqtr3 2492 |
. . . . . . . 8
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7 | 5, 6 | sylan 479 |
. . . . . . 7
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8 | rncoeq 5104 |
. . . . . . . . 9
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9 | 8 | eqeq1d 2473 |
. . . . . . . 8
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10 | 9 | biimpar 493 |
. . . . . . 7
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11 | 7, 10 | sylan 479 |
. . . . . 6
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12 | 11 | an32s 821 |
. . . . 5
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13 | 12 | adantrl 730 |
. . . 4
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14 | 4, 13 | jca 541 |
. . 3
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15 | 1, 2, 14 | syl2anb 487 |
. 2
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16 | dffo2 5810 |
. 2
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17 | 15, 16 | sylibr 217 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-fun 5591 df-fn 5592 df-f 5593 df-fo 5595 |
This theorem is referenced by: f1oco 5850 wdomtr 8108 fin1a2lem7 8854 cofull 15917 uniiccdif 22614 |
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