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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 990 | . . . 4 | |
2 | simp2l 992 | . . . 4 | |
3 | funsng 5139 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 408 | . . 3 |
5 | simp1r 991 | . . . 4 | |
6 | simp2r 993 | . . . 4 | |
7 | funsng 5139 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 408 | . . 3 |
9 | dmsnopg 4980 | . . . . . 6 | |
10 | 2, 9 | syl 14 | . . . . 5 |
11 | dmsnopg 4980 | . . . . . 6 | |
12 | 6, 11 | syl 14 | . . . . 5 |
13 | 10, 12 | ineq12d 3248 | . . . 4 |
14 | disjsn2 3556 | . . . . 5 | |
15 | 14 | 3ad2ant3 989 | . . . 4 |
16 | 13, 15 | eqtrd 2150 | . . 3 |
17 | funun 5137 | . . 3 | |
18 | 4, 8, 16, 17 | syl21anc 1200 | . 2 |
19 | df-pr 3504 | . . 3 | |
20 | 19 | funeqi 5114 | . 2 |
21 | 18, 20 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wne 2285 cun 3039 cin 3040 c0 3333 csn 3497 cpr 3498 cop 3500 cdm 4509 wfun 5087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-fun 5095 |
This theorem is referenced by: funtpg 5144 funpr 5145 fnprg 5148 2strbasg 11987 2stropg 11988 |
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