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Mirrors > Home > MPE Home > Th. List > xpdom1g | Structured version 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 7419 |
. . . 4
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2 | 1 | brrelexi 4980 |
. . 3
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3 | xpcomeng 7506 |
. . . 4
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4 | 3 | ancoms 453 |
. . 3
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5 | 2, 4 | sylan2 474 |
. 2
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6 | xpdom2g 7510 |
. . 3
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7 | 1 | brrelex2i 4981 |
. . . 4
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8 | xpcomeng 7506 |
. . . 4
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9 | 7, 8 | sylan2 474 |
. . 3
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10 | domentr 7471 |
. . 3
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11 | 6, 9, 10 | syl2anc 661 |
. 2
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12 | endomtr 7470 |
. 2
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13 | 5, 11, 12 | syl2anc 661 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-br 4394 df-opab 4452 df-mpt 4453 df-id 4737 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-1st 6680 df-2nd 6681 df-en 7414 df-dom 7415 |
This theorem is referenced by: xpdom1 7513 xpen 7577 infpwfien 8336 iunctb 8842 canthp1lem1 8923 gchxpidm 8940 xpct 26154 fnct 26157 |
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