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Mirrors > Home > ILE Home > Th. List > dfoprab3 | Unicode version |
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
Ref | Expression |
---|---|
dfoprab3.1 |
Ref | Expression |
---|---|
dfoprab3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfoprab3s 6150 | . 2 | |
2 | vex 2724 | . . . . . 6 | |
3 | 1stexg 6127 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | 2ndexg 6128 | . . . . . 6 | |
6 | 2, 5 | ax-mp 5 | . . . . 5 |
7 | eqcom 2166 | . . . . . . . . . 10 | |
8 | eqcom 2166 | . . . . . . . . . 10 | |
9 | 7, 8 | anbi12i 456 | . . . . . . . . 9 |
10 | eqopi 6132 | . . . . . . . . 9 | |
11 | 9, 10 | sylan2b 285 | . . . . . . . 8 |
12 | dfoprab3.1 | . . . . . . . 8 | |
13 | 11, 12 | syl 14 | . . . . . . 7 |
14 | 13 | bicomd 140 | . . . . . 6 |
15 | 14 | ex 114 | . . . . 5 |
16 | 4, 6, 15 | sbc2iedv 3018 | . . . 4 |
17 | 16 | pm5.32i 450 | . . 3 |
18 | 17 | opabbii 4043 | . 2 |
19 | 1, 18 | eqtr2i 2186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 wsbc 2946 cop 3573 copab 4036 cxp 4596 cfv 5182 coprab 5837 c1st 6098 c2nd 6099 |
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-v 2723 df-sbc 2947 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-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-oprab 5840 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: dfoprab4 6152 df1st2 6178 df2nd2 6179 |
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