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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 2081 | . . . . . 6 | |
2 | 1 | albii 1450 | . . . . 5 |
3 | funopab 5205 | . . . . 5 | |
4 | df-ral 2440 | . . . . 5 | |
5 | 2, 3, 4 | 3bitr4ri 212 | . . . 4 |
6 | dmopab3 4799 | . . . 4 | |
7 | 5, 6 | anbi12i 456 | . . 3 |
8 | r19.26 2583 | . . 3 | |
9 | df-fn 5173 | . . 3 | |
10 | 7, 8, 9 | 3bitr4i 211 | . 2 |
11 | eu5 2053 | . . . 4 | |
12 | ancom 264 | . . . 4 | |
13 | 11, 12 | bitri 183 | . . 3 |
14 | 13 | ralbii 2463 | . 2 |
15 | fnopabg.1 | . . 3 | |
16 | 15 | fneq1i 5264 | . 2 |
17 | 10, 14, 16 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wceq 1335 wex 1472 weu 2006 wmo 2007 wcel 2128 wral 2435 copab 4024 cdm 4586 wfun 5164 wfn 5165 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-fun 5172 df-fn 5173 |
This theorem is referenced by: fnopab 5294 mptfng 5295 |
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