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Mirrors > Home > ILE Home > Th. List > fnfi | Unicode version |
Description: A version of fnex 5689 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
fnfi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresdm 5279 | . . 3 | |
2 | 1 | adantr 274 | . 2 |
3 | reseq2 4861 | . . . 4 | |
4 | 3 | eleq1d 2226 | . . 3 |
5 | reseq2 4861 | . . . 4 | |
6 | 5 | eleq1d 2226 | . . 3 |
7 | reseq2 4861 | . . . 4 | |
8 | 7 | eleq1d 2226 | . . 3 |
9 | reseq2 4861 | . . . 4 | |
10 | 9 | eleq1d 2226 | . . 3 |
11 | res0 4870 | . . . . 5 | |
12 | 0fin 6829 | . . . . 5 | |
13 | 11, 12 | eqeltri 2230 | . . . 4 |
14 | 13 | a1i 9 | . . 3 |
15 | resundi 4879 | . . . . 5 | |
16 | simp-4l 531 | . . . . . . . 8 | |
17 | simplrr 526 | . . . . . . . . 9 | |
18 | 17 | eldifad 3113 | . . . . . . . 8 |
19 | fnressn 5653 | . . . . . . . 8 | |
20 | 16, 18, 19 | syl2anc 409 | . . . . . . 7 |
21 | 20 | uneq2d 3261 | . . . . . 6 |
22 | simpr 109 | . . . . . . 7 | |
23 | 17 | elexd 2725 | . . . . . . . 8 |
24 | funfvex 5485 | . . . . . . . . . 10 | |
25 | 24 | funfni 5270 | . . . . . . . . 9 |
26 | 16, 18, 25 | syl2anc 409 | . . . . . . . 8 |
27 | opexg 4188 | . . . . . . . 8 | |
28 | 23, 26, 27 | syl2anc 409 | . . . . . . 7 |
29 | 17 | eldifbd 3114 | . . . . . . . 8 |
30 | opeldmg 4791 | . . . . . . . . . . 11 | |
31 | 18, 26, 30 | syl2anc 409 | . . . . . . . . . 10 |
32 | dmres 4887 | . . . . . . . . . . 11 | |
33 | 32 | eleq2i 2224 | . . . . . . . . . 10 |
34 | 31, 33 | syl6ib 160 | . . . . . . . . 9 |
35 | elin 3290 | . . . . . . . . . 10 | |
36 | 35 | simplbi 272 | . . . . . . . . 9 |
37 | 34, 36 | syl6 33 | . . . . . . . 8 |
38 | 29, 37 | mtod 653 | . . . . . . 7 |
39 | unsnfi 6863 | . . . . . . 7 | |
40 | 22, 28, 38, 39 | syl3anc 1220 | . . . . . 6 |
41 | 21, 40 | eqeltrd 2234 | . . . . 5 |
42 | 15, 41 | eqeltrid 2244 | . . . 4 |
43 | 42 | ex 114 | . . 3 |
44 | simpr 109 | . . 3 | |
45 | 4, 6, 8, 10, 14, 43, 44 | findcard2sd 6837 | . 2 |
46 | 2, 45 | eqeltrrd 2235 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1335 wcel 2128 cvv 2712 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3394 csn 3560 cop 3563 cdm 4586 cres 4588 wfn 5165 cfv 5170 cfn 6685 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1o 6363 df-er 6480 df-en 6686 df-fin 6688 |
This theorem is referenced by: fundmfibi 6883 resfnfinfinss 6884 fihashf1rn 10663 fihashfn 10674 |
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