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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 6725 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | bren 6725 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antlr 486 | . . 3 |
7 | relen 6722 | . . . . . . 7 | |
8 | 7 | brrelex1i 4654 | . . . . . 6 |
9 | 7 | brrelex1i 4654 | . . . . . 6 |
10 | xpexg 4725 | . . . . . 6 | |
11 | 8, 9, 10 | syl2an 287 | . . . . 5 |
12 | 11 | ad2antrr 485 | . . . 4 |
13 | simplr 525 | . . . . . 6 | |
14 | f1ofn 5443 | . . . . . . . 8 | |
15 | dffn5im 5542 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | f1oeq1 5431 | . . . . . . 7 | |
18 | 13, 16, 17 | 3syl 17 | . . . . . 6 |
19 | 13, 18 | mpbid 146 | . . . . 5 |
20 | simpr 109 | . . . . . 6 | |
21 | f1ofn 5443 | . . . . . . . 8 | |
22 | dffn5im 5542 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | f1oeq1 5431 | . . . . . . 7 | |
25 | 20, 23, 24 | 3syl 17 | . . . . . 6 |
26 | 20, 25 | mpbid 146 | . . . . 5 |
27 | 19, 26 | xpf1o 6822 | . . . 4 |
28 | f1oeng 6735 | . . . 4 | |
29 | 12, 27, 28 | syl2anc 409 | . . 3 |
30 | 6, 29 | exlimddv 1891 | . 2 |
31 | 3, 30 | exlimddv 1891 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 cvv 2730 cop 3586 class class class wbr 3989 cmpt 4050 cxp 4609 wfn 5193 wf1o 5197 cfv 5198 cmpo 5855 cen 6716 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-en 6719 |
This theorem is referenced by: xpdjuen 7195 xpnnen 12349 xpomen 12350 qnnen 12386 |
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