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Mirrors > Home > ILE Home > Th. List > brdomg | Unicode version |
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
brdomg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6719 | . . . 4 | |
2 | 1 | brrelex1i 4652 | . . 3 |
3 | 2 | a1i 9 | . 2 |
4 | f1f 5401 | . . . . 5 | |
5 | fdm 5351 | . . . . . 6 | |
6 | vex 2733 | . . . . . . 7 | |
7 | 6 | dmex 4875 | . . . . . 6 |
8 | 5, 7 | eqeltrrdi 2262 | . . . . 5 |
9 | 4, 8 | syl 14 | . . . 4 |
10 | 9 | exlimiv 1591 | . . 3 |
11 | 10 | a1i 9 | . 2 |
12 | f1eq2 5397 | . . . . 5 | |
13 | 12 | exbidv 1818 | . . . 4 |
14 | f1eq3 5398 | . . . . 5 | |
15 | 14 | exbidv 1818 | . . . 4 |
16 | df-dom 6716 | . . . 4 | |
17 | 13, 15, 16 | brabg 4252 | . . 3 |
18 | 17 | expcom 115 | . 2 |
19 | 3, 11, 18 | pm5.21ndd 700 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1348 wex 1485 wcel 2141 cvv 2730 class class class wbr 3987 cdm 4609 wf 5192 wf1 5193 cdom 6713 |
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-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
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-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-dm 4619 df-rn 4620 df-fn 5199 df-f 5200 df-f1 5201 df-dom 6716 |
This theorem is referenced by: brdomi 6723 brdom 6724 f1dom2g 6730 f1domg 6732 dom3d 6748 phplem4dom 6836 djudom 7066 difinfsn 7073 djudoml 7183 djudomr 7184 nninfdc 12395 |
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