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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 6641 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | bren 6641 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antlr 480 | . . 3 |
7 | relen 6638 | . . . . . . 7 | |
8 | 7 | brrelex1i 4582 | . . . . . 6 |
9 | 7 | brrelex1i 4582 | . . . . . 6 |
10 | xpexg 4653 | . . . . . 6 | |
11 | 8, 9, 10 | syl2an 287 | . . . . 5 |
12 | 11 | ad2antrr 479 | . . . 4 |
13 | simplr 519 | . . . . . 6 | |
14 | f1ofn 5368 | . . . . . . . 8 | |
15 | dffn5im 5467 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | f1oeq1 5356 | . . . . . . 7 | |
18 | 13, 16, 17 | 3syl 17 | . . . . . 6 |
19 | 13, 18 | mpbid 146 | . . . . 5 |
20 | simpr 109 | . . . . . 6 | |
21 | f1ofn 5368 | . . . . . . . 8 | |
22 | dffn5im 5467 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | f1oeq1 5356 | . . . . . . 7 | |
25 | 20, 23, 24 | 3syl 17 | . . . . . 6 |
26 | 20, 25 | mpbid 146 | . . . . 5 |
27 | 19, 26 | xpf1o 6738 | . . . 4 |
28 | f1oeng 6651 | . . . 4 | |
29 | 12, 27, 28 | syl2anc 408 | . . 3 |
30 | 6, 29 | exlimddv 1870 | . 2 |
31 | 3, 30 | exlimddv 1870 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2686 cop 3530 class class class wbr 3929 cmpt 3989 cxp 4537 wfn 5118 wf1o 5122 cfv 5123 cmpo 5776 cen 6632 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
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-reu 2423 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-iun 3815 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-fo 5129 df-f1o 5130 df-fv 5131 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-en 6635 |
This theorem is referenced by: xpdjuen 7074 xpnnen 11907 xpomen 11908 qnnen 11944 |
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