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Mirrors > Home > ILE Home > Th. List > ovprc | Unicode version |
Description: The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ovprc1.1 |
Ref | Expression |
---|---|
ovprc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5840 | . 2 | |
2 | opprc 3774 | . . . 4 | |
3 | 0ex 4104 | . . . 4 | |
4 | 2, 3 | eqeltrdi 2255 | . . 3 |
5 | df-br 3978 | . . . . 5 | |
6 | ovprc1.1 | . . . . . 6 | |
7 | brrelex12 4637 | . . . . . 6 | |
8 | 6, 7 | mpan 421 | . . . . 5 |
9 | 5, 8 | sylbir 134 | . . . 4 |
10 | 9 | con3i 622 | . . 3 |
11 | ndmfvg 5512 | . . 3 | |
12 | 4, 10, 11 | syl2anc 409 | . 2 |
13 | 1, 12 | syl5eq 2209 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1342 wcel 2135 cvv 2722 c0 3405 cop 3574 class class class wbr 3977 cdm 4599 wrel 4604 cfv 5183 (class class class)co 5837 |
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-in1 604 ax-in2 605 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-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-xp 4605 df-rel 4606 df-dm 4609 df-iota 5148 df-fv 5191 df-ov 5840 |
This theorem is referenced by: ovprc1 5870 ovprc2 5871 |
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