Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fnopabg | Structured version Visualization version 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 2531 | . . . . . 6 | |
2 | 1 | albii 1747 | . . . . 5 |
3 | funopab 5923 | . . . . 5 | |
4 | df-ral 2917 | . . . . 5 | |
5 | 2, 3, 4 | 3bitr4ri 293 | . . . 4 |
6 | dmopab3 5337 | . . . 4 | |
7 | 5, 6 | anbi12i 733 | . . 3 |
8 | r19.26 3064 | . . 3 | |
9 | df-fn 5891 | . . 3 | |
10 | 7, 8, 9 | 3bitr4i 292 | . 2 |
11 | eu5 2496 | . . . 4 | |
12 | ancom 466 | . . . 4 | |
13 | 11, 12 | bitri 264 | . . 3 |
14 | 13 | ralbii 2980 | . 2 |
15 | fnopabg.1 | . . 3 | |
16 | 15 | fneq1i 5985 | . 2 |
17 | 10, 14, 16 | 3bitr4i 292 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 weu 2470 wmo 2471 wral 2912 copab 4712 cdm 5114 wfun 5882 wfn 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-fun 5890 df-fn 5891 |
This theorem is referenced by: fnopab 6018 mptfng 6019 axcontlem2 25845 |
Copyright terms: Public domain | W3C validator |