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Mirrors > Home > ILE Home > Th. List > f1mpt | Unicode version |
Description: Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
f1mpt.1 | |
f1mpt.2 |
Ref | Expression |
---|---|
f1mpt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1mpt.1 | . . . 4 | |
2 | nfmpt1 4021 | . . . 4 | |
3 | 1, 2 | nfcxfr 2278 | . . 3 |
4 | nfcv 2281 | . . 3 | |
5 | 3, 4 | dff13f 5671 | . 2 |
6 | 1 | fmpt 5570 | . . 3 |
7 | 6 | anbi1i 453 | . 2 |
8 | f1mpt.2 | . . . . . . 7 | |
9 | 8 | eleq1d 2208 | . . . . . 6 |
10 | 9 | cbvralv 2654 | . . . . 5 |
11 | raaanv 3470 | . . . . . 6 | |
12 | 1 | fvmpt2 5504 | . . . . . . . . . . . . . 14 |
13 | 8, 1 | fvmptg 5497 | . . . . . . . . . . . . . 14 |
14 | 12, 13 | eqeqan12d 2155 | . . . . . . . . . . . . 13 |
15 | 14 | an4s 577 | . . . . . . . . . . . 12 |
16 | 15 | imbi1d 230 | . . . . . . . . . . 11 |
17 | 16 | ex 114 | . . . . . . . . . 10 |
18 | 17 | ralimdva 2499 | . . . . . . . . 9 |
19 | ralbi 2564 | . . . . . . . . 9 | |
20 | 18, 19 | syl6 33 | . . . . . . . 8 |
21 | 20 | ralimia 2493 | . . . . . . 7 |
22 | ralbi 2564 | . . . . . . 7 | |
23 | 21, 22 | syl 14 | . . . . . 6 |
24 | 11, 23 | sylbir 134 | . . . . 5 |
25 | 10, 24 | sylan2b 285 | . . . 4 |
26 | 25 | anidms 394 | . . 3 |
27 | 26 | pm5.32i 449 | . 2 |
28 | 5, 7, 27 | 3bitr2i 207 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 cmpt 3989 wf 5119 wf1 5120 cfv 5123 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fv 5131 |
This theorem is referenced by: 1domsn 6713 difinfsnlem 6984 |
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