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Mirrors > Home > ILE Home > Th. List > funprg | Unicode version |
Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
Ref | Expression |
---|---|
funprg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1021 |
. . . 4
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2 | simp2l 1023 |
. . . 4
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3 | funsng 5258 |
. . . 4
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4 | 1, 2, 3 | syl2anc 411 |
. . 3
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5 | simp1r 1022 |
. . . 4
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6 | simp2r 1024 |
. . . 4
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7 | funsng 5258 |
. . . 4
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8 | 5, 6, 7 | syl2anc 411 |
. . 3
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9 | dmsnopg 5096 |
. . . . . 6
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10 | 2, 9 | syl 14 |
. . . . 5
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11 | dmsnopg 5096 |
. . . . . 6
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12 | 6, 11 | syl 14 |
. . . . 5
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13 | 10, 12 | ineq12d 3337 |
. . . 4
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14 | disjsn2 3654 |
. . . . 5
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15 | 14 | 3ad2ant3 1020 |
. . . 4
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16 | 13, 15 | eqtrd 2210 |
. . 3
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17 | funun 5256 |
. . 3
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18 | 4, 8, 16, 17 | syl21anc 1237 |
. 2
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19 | df-pr 3598 |
. . 3
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20 | 19 | funeqi 5233 |
. 2
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21 | 18, 20 | sylibr 134 |
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-in1 614 ax-in2 615 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 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-fun 5214 |
This theorem is referenced by: funtpg 5263 funpr 5264 fnprg 5267 2strbasg 12554 2stropg 12555 |
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