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Mirrors > Home > ILE Home > Th. List > foco2 | Unicode version |
Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
foco2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 997 |
. 2
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2 | foelrn 5753 |
. . . . . 6
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3 | ffvelcdm 5649 |
. . . . . . . . . 10
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4 | 3 | adantll 476 |
. . . . . . . . 9
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5 | fvco3 5587 |
. . . . . . . . . 10
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6 | 5 | adantll 476 |
. . . . . . . . 9
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7 | fveq2 5515 |
. . . . . . . . . . 11
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8 | 7 | eqeq2d 2189 |
. . . . . . . . . 10
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9 | 8 | rspcev 2841 |
. . . . . . . . 9
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10 | 4, 6, 9 | syl2anc 411 |
. . . . . . . 8
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11 | eqeq1 2184 |
. . . . . . . . 9
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12 | 11 | rexbidv 2478 |
. . . . . . . 8
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13 | 10, 12 | syl5ibrcom 157 |
. . . . . . 7
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14 | 13 | rexlimdva 2594 |
. . . . . 6
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15 | 2, 14 | syl5 32 |
. . . . 5
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16 | 15 | impl 380 |
. . . 4
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17 | 16 | ralrimiva 2550 |
. . 3
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18 | 17 | 3impa 1194 |
. 2
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19 | dffo3 5663 |
. 2
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20 | 1, 18, 19 | sylanbrc 417 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 df-fv 5224 |
This theorem is referenced by: (None) |
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