Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2nd2val | Structured version Visualization version Unicode version |
Description: Value of an alternate definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
2nd2val |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elvv 5177 | . . 3 | |
2 | fveq2 6191 | . . . . . 6 | |
3 | df-ov 6653 | . . . . . . 7 | |
4 | vex 3203 | . . . . . . . 8 | |
5 | vex 3203 | . . . . . . . 8 | |
6 | simpr 477 | . . . . . . . . 9 | |
7 | mpt2v 6750 | . . . . . . . . . 10 | |
8 | 7 | eqcomi 2631 | . . . . . . . . 9 |
9 | 6, 8, 5 | ovmpt2a 6791 | . . . . . . . 8 |
10 | 4, 5, 9 | mp2an 708 | . . . . . . 7 |
11 | 3, 10 | eqtr3i 2646 | . . . . . 6 |
12 | 2, 11 | syl6eq 2672 | . . . . 5 |
13 | 4, 5 | op2ndd 7179 | . . . . 5 |
14 | 12, 13 | eqtr4d 2659 | . . . 4 |
15 | 14 | exlimivv 1860 | . . 3 |
16 | 1, 15 | sylbi 207 | . 2 |
17 | vex 3203 | . . . . . . . . . 10 | |
18 | vex 3203 | . . . . . . . . . 10 | |
19 | 17, 18 | pm3.2i 471 | . . . . . . . . 9 |
20 | ax6ev 1890 | . . . . . . . . 9 | |
21 | 19, 20 | 2th 254 | . . . . . . . 8 |
22 | 21 | opabbii 4717 | . . . . . . 7 |
23 | df-xp 5120 | . . . . . . 7 | |
24 | dmoprab 6741 | . . . . . . 7 | |
25 | 22, 23, 24 | 3eqtr4ri 2655 | . . . . . 6 |
26 | 25 | eleq2i 2693 | . . . . 5 |
27 | ndmfv 6218 | . . . . 5 | |
28 | 26, 27 | sylnbir 321 | . . . 4 |
29 | rnsnn0 5601 | . . . . . . . 8 | |
30 | 29 | biimpri 218 | . . . . . . 7 |
31 | 30 | necon1bi 2822 | . . . . . 6 |
32 | 31 | unieqd 4446 | . . . . 5 |
33 | uni0 4465 | . . . . 5 | |
34 | 32, 33 | syl6eq 2672 | . . . 4 |
35 | 28, 34 | eqtr4d 2659 | . . 3 |
36 | 2ndval 7171 | . . 3 | |
37 | 35, 36 | syl6eqr 2674 | . 2 |
38 | 16, 37 | pm2.61i 176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wa 384 wceq 1483 wex 1704 wcel 1990 wne 2794 cvv 3200 c0 3915 csn 4177 cop 4183 cuni 4436 copab 4712 cxp 5112 cdm 5114 crn 5115 cfv 5888 (class class class)co 6650 coprab 6651 cmpt2 6652 c2nd 7167 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
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-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-2nd 7169 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |