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Mirrors > Home > MPE Home > Th. List > ov2gf | Structured version Visualization version Unicode version |
Description: The value of an operation class abstraction. A version of ovmpt2g 6795 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ov2gf.a | |
ov2gf.c | |
ov2gf.d | |
ov2gf.1 | |
ov2gf.2 | |
ov2gf.3 | |
ov2gf.4 | |
ov2gf.5 |
Ref | Expression |
---|---|
ov2gf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3212 | . . 3 | |
2 | ov2gf.a | . . . 4 | |
3 | ov2gf.c | . . . 4 | |
4 | ov2gf.d | . . . 4 | |
5 | ov2gf.1 | . . . . . 6 | |
6 | 5 | nfel1 2779 | . . . . 5 |
7 | ov2gf.5 | . . . . . . . 8 | |
8 | nfmpt21 6722 | . . . . . . . 8 | |
9 | 7, 8 | nfcxfr 2762 | . . . . . . 7 |
10 | nfcv 2764 | . . . . . . 7 | |
11 | 2, 9, 10 | nfov 6676 | . . . . . 6 |
12 | 11, 5 | nfeq 2776 | . . . . 5 |
13 | 6, 12 | nfim 1825 | . . . 4 |
14 | ov2gf.2 | . . . . . 6 | |
15 | 14 | nfel1 2779 | . . . . 5 |
16 | nfmpt22 6723 | . . . . . . . 8 | |
17 | 7, 16 | nfcxfr 2762 | . . . . . . 7 |
18 | 3, 17, 4 | nfov 6676 | . . . . . 6 |
19 | 18, 14 | nfeq 2776 | . . . . 5 |
20 | 15, 19 | nfim 1825 | . . . 4 |
21 | ov2gf.3 | . . . . . 6 | |
22 | 21 | eleq1d 2686 | . . . . 5 |
23 | oveq1 6657 | . . . . . 6 | |
24 | 23, 21 | eqeq12d 2637 | . . . . 5 |
25 | 22, 24 | imbi12d 334 | . . . 4 |
26 | ov2gf.4 | . . . . . 6 | |
27 | 26 | eleq1d 2686 | . . . . 5 |
28 | oveq2 6658 | . . . . . 6 | |
29 | 28, 26 | eqeq12d 2637 | . . . . 5 |
30 | 27, 29 | imbi12d 334 | . . . 4 |
31 | 7 | ovmpt4g 6783 | . . . . 5 |
32 | 31 | 3expia 1267 | . . . 4 |
33 | 2, 3, 4, 13, 20, 25, 30, 32 | vtocl2gaf 3273 | . . 3 |
34 | 1, 33 | syl5 34 | . 2 |
35 | 34 | 3impia 1261 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wnfc 2751 cvv 3200 (class class class)co 6650 cmpt2 6652 |
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-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-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: (None) |
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