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Mirrors > Home > MPE Home > Th. List > wdomen2 | Structured version Visualization version Unicode version |
Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
wdomen2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 |
. . 3
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2 | endom 7601 |
. . . 4
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3 | domwdom 8094 |
. . . 4
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4 | 2, 3 | syl 17 |
. . 3
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5 | wdomtr 8095 |
. . 3
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6 | 1, 4, 5 | syl2anr 481 |
. 2
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7 | id 22 |
. . 3
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8 | ensym 7623 |
. . . 4
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9 | endom 7601 |
. . . 4
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10 | domwdom 8094 |
. . . 4
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11 | 8, 9, 10 | 3syl 18 |
. . 3
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12 | wdomtr 8095 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 7, 11, 12 | syl2anr 481 |
. 2
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14 | 6, 13 | impbida 844 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-er 7368 df-en 7575 df-dom 7576 df-sdom 7577 df-wdom 8079 |
This theorem is referenced by: (None) |
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