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Mirrors > Home > MPE Home > Th. List > fcofo | Structured version Visualization version Unicode version |
Description: An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
fcofo |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1009 |
. 2
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2 | ffvelrn 6025 |
. . . . 5
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3 | 2 | 3ad2antl2 1172 |
. . . 4
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4 | simpl3 1014 |
. . . . . 6
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5 | 4 | fveq1d 5872 |
. . . . 5
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6 | fvco3 5947 |
. . . . . 6
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7 | 6 | 3ad2antl2 1172 |
. . . . 5
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8 | fvresi 6095 |
. . . . . 6
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9 | 8 | adantl 468 |
. . . . 5
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10 | 5, 7, 9 | 3eqtr3rd 2496 |
. . . 4
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11 | fveq2 5870 |
. . . . . 6
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12 | 11 | eqeq2d 2463 |
. . . . 5
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13 | 12 | rspcev 3152 |
. . . 4
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14 | 3, 10, 13 | syl2anc 667 |
. . 3
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15 | 14 | ralrimiva 2804 |
. 2
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16 | dffo3 6042 |
. 2
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17 | 1, 15, 16 | sylanbrc 671 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-fo 5591 df-fv 5593 |
This theorem is referenced by: fcof1od 6197 |
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