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Mirrors > Home > MPE Home > Th. List > fvopab5 | Structured version Visualization version Unicode version |
Description: The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fvopab5.1 | |
fvopab5.2 |
Ref | Expression |
---|---|
fvopab5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3212 | . 2 | |
2 | df-fv 5896 | . . . 4 | |
3 | breq2 4657 | . . . . 5 | |
4 | nfcv 2764 | . . . . . 6 | |
5 | fvopab5.1 | . . . . . . 7 | |
6 | nfopab2 4720 | . . . . . . 7 | |
7 | 5, 6 | nfcxfr 2762 | . . . . . 6 |
8 | nfcv 2764 | . . . . . 6 | |
9 | 4, 7, 8 | nfbr 4699 | . . . . 5 |
10 | nfv 1843 | . . . . 5 | |
11 | 3, 9, 10 | cbviota 5856 | . . . 4 |
12 | 2, 11 | eqtri 2644 | . . 3 |
13 | nfcv 2764 | . . . . . . 7 | |
14 | nfopab1 4719 | . . . . . . . 8 | |
15 | 5, 14 | nfcxfr 2762 | . . . . . . 7 |
16 | nfcv 2764 | . . . . . . 7 | |
17 | 13, 15, 16 | nfbr 4699 | . . . . . 6 |
18 | nfv 1843 | . . . . . 6 | |
19 | 17, 18 | nfbi 1833 | . . . . 5 |
20 | breq1 4656 | . . . . . 6 | |
21 | fvopab5.2 | . . . . . 6 | |
22 | 20, 21 | bibi12d 335 | . . . . 5 |
23 | df-br 4654 | . . . . . 6 | |
24 | 5 | eleq2i 2693 | . . . . . 6 |
25 | opabid 4982 | . . . . . 6 | |
26 | 23, 24, 25 | 3bitri 286 | . . . . 5 |
27 | 19, 22, 26 | vtoclg1f 3265 | . . . 4 |
28 | 27 | iotabidv 5872 | . . 3 |
29 | 12, 28 | syl5eq 2668 | . 2 |
30 | 1, 29 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wcel 1990 cvv 3200 cop 4183 class class class wbr 4653 copab 4712 cio 5849 cfv 5888 |
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-rex 2918 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-uni 4437 df-br 4654 df-opab 4713 df-iota 5851 df-fv 5896 |
This theorem is referenced by: ajval 27717 adjval 28749 |
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