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Mirrors > Home > ILE Home > Th. List > fifo | Unicode version |
Description: Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
fifo.1 |
Ref | Expression |
---|---|
fifo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsni 3700 | . . . . . . 7 | |
2 | eldifi 3240 | . . . . . . . . 9 | |
3 | 2 | elin2d 3308 | . . . . . . . 8 |
4 | fin0 6843 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 1, 5 | mpbid 146 | . . . . . 6 |
7 | inteximm 4123 | . . . . . 6 | |
8 | 6, 7 | syl 14 | . . . . 5 |
9 | 8 | rgen 2517 | . . . 4 |
10 | fifo.1 | . . . . 5 | |
11 | 10 | fnmpt 5309 | . . . 4 |
12 | 9, 11 | mp1i 10 | . . 3 |
13 | dffn4 5411 | . . 3 | |
14 | 12, 13 | sylib 121 | . 2 |
15 | elfi2 6929 | . . . . 5 | |
16 | 10 | elrnmpt 4848 | . . . . . 6 |
17 | 16 | elv 2726 | . . . . 5 |
18 | 15, 17 | bitr4di 197 | . . . 4 |
19 | 18 | eqrdv 2162 | . . 3 |
20 | foeq3 5403 | . . 3 | |
21 | 19, 20 | syl 14 | . 2 |
22 | 14, 21 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1342 wex 1479 wcel 2135 wne 2334 wral 2442 wrex 2443 cvv 2722 cdif 3109 cin 3111 c0 3405 cpw 3554 csn 3571 cint 3819 cmpt 4038 crn 4600 wfn 5178 wfo 5181 cfv 5183 cfn 6698 cfi 6925 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-er 6493 df-en 6699 df-fin 6701 df-fi 6926 |
This theorem is referenced by: (None) |
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