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Mirrors > Home > ILE Home > Th. List > xpdom1g | Unicode version |
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
xpdom1g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6702 | . . . 4 | |
2 | 1 | brrelex1i 4641 | . . 3 |
3 | xpcomeng 6785 | . . . 4 | |
4 | 3 | ancoms 266 | . . 3 |
5 | 2, 4 | sylan2 284 | . 2 |
6 | xpdom2g 6789 | . . 3 | |
7 | 1 | brrelex2i 4642 | . . . 4 |
8 | xpcomeng 6785 | . . . 4 | |
9 | 7, 8 | sylan2 284 | . . 3 |
10 | domentr 6748 | . . 3 | |
11 | 6, 9, 10 | syl2anc 409 | . 2 |
12 | endomtr 6747 | . 2 | |
13 | 5, 11, 12 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 cvv 2721 class class class wbr 3976 cxp 4596 cen 6695 cdom 6696 |
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 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 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 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-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-en 6698 df-dom 6699 |
This theorem is referenced by: xpdom1 6792 xpct 12272 |
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