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Mirrors > Home > ILE Home > Th. List > xpen | Unicode version |
Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) |
Ref | Expression |
---|---|
xpen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6609 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | bren 6609 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antlr 480 | . . 3 |
7 | relen 6606 | . . . . . . 7 | |
8 | 7 | brrelex1i 4552 | . . . . . 6 |
9 | 7 | brrelex1i 4552 | . . . . . 6 |
10 | xpexg 4623 | . . . . . 6 | |
11 | 8, 9, 10 | syl2an 287 | . . . . 5 |
12 | 11 | ad2antrr 479 | . . . 4 |
13 | simplr 504 | . . . . . 6 | |
14 | f1ofn 5336 | . . . . . . . 8 | |
15 | dffn5im 5435 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | f1oeq1 5326 | . . . . . . 7 | |
18 | 13, 16, 17 | 3syl 17 | . . . . . 6 |
19 | 13, 18 | mpbid 146 | . . . . 5 |
20 | simpr 109 | . . . . . 6 | |
21 | f1ofn 5336 | . . . . . . . 8 | |
22 | dffn5im 5435 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | f1oeq1 5326 | . . . . . . 7 | |
25 | 20, 23, 24 | 3syl 17 | . . . . . 6 |
26 | 20, 25 | mpbid 146 | . . . . 5 |
27 | 19, 26 | xpf1o 6706 | . . . 4 |
28 | f1oeng 6619 | . . . 4 | |
29 | 12, 27, 28 | syl2anc 408 | . . 3 |
30 | 6, 29 | exlimddv 1854 | . 2 |
31 | 3, 30 | exlimddv 1854 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cvv 2660 cop 3500 class class class wbr 3899 cmpt 3959 cxp 4507 wfn 5088 wf1o 5092 cfv 5093 cmpo 5744 cen 6600 |
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 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-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-en 6603 |
This theorem is referenced by: xpdjuen 7042 xpnnen 11834 xpomen 11835 qnnen 11871 |
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