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Mirrors > Home > ILE Home > Th. List > fnopabg | Unicode version |
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) |
Ref | Expression |
---|---|
fnopabg.1 |
Ref | Expression |
---|---|
fnopabg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moanimv 2074 | . . . . . 6 | |
2 | 1 | albii 1446 | . . . . 5 |
3 | funopab 5158 | . . . . 5 | |
4 | df-ral 2421 | . . . . 5 | |
5 | 2, 3, 4 | 3bitr4ri 212 | . . . 4 |
6 | dmopab3 4752 | . . . 4 | |
7 | 5, 6 | anbi12i 455 | . . 3 |
8 | r19.26 2558 | . . 3 | |
9 | df-fn 5126 | . . 3 | |
10 | 7, 8, 9 | 3bitr4i 211 | . 2 |
11 | eu5 2046 | . . . 4 | |
12 | ancom 264 | . . . 4 | |
13 | 11, 12 | bitri 183 | . . 3 |
14 | 13 | ralbii 2441 | . 2 |
15 | fnopabg.1 | . . 3 | |
16 | 15 | fneq1i 5217 | . 2 |
17 | 10, 14, 16 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wex 1468 wcel 1480 weu 1999 wmo 2000 wral 2416 copab 3988 cdm 4539 wfun 5117 wfn 5118 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-fun 5125 df-fn 5126 |
This theorem is referenced by: fnopab 5247 mptfng 5248 |
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