Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > off2 | Structured version Visualization version Unicode version |
Description: The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.) |
Ref | Expression |
---|---|
off2.1 | |
off2.2 | |
off2.3 | |
off2.4 | |
off2.5 | |
off2.6 |
Ref | Expression |
---|---|
off2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | off2.2 | . . . . . 6 | |
2 | 1 | adantr 481 | . . . . 5 |
3 | off2.6 | . . . . . . 7 | |
4 | inss1 3833 | . . . . . . 7 | |
5 | 3, 4 | syl6eqssr 3656 | . . . . . 6 |
6 | 5 | sselda 3603 | . . . . 5 |
7 | 2, 6 | ffvelrnd 6360 | . . . 4 |
8 | off2.3 | . . . . . 6 | |
9 | 8 | adantr 481 | . . . . 5 |
10 | inss2 3834 | . . . . . . 7 | |
11 | 3, 10 | syl6eqssr 3656 | . . . . . 6 |
12 | 11 | sselda 3603 | . . . . 5 |
13 | 9, 12 | ffvelrnd 6360 | . . . 4 |
14 | off2.1 | . . . . . 6 | |
15 | 14 | ralrimivva 2971 | . . . . 5 |
16 | 15 | adantr 481 | . . . 4 |
17 | ovrspc2v 6672 | . . . 4 | |
18 | 7, 13, 16, 17 | syl21anc 1325 | . . 3 |
19 | eqid 2622 | . . 3 | |
20 | 18, 19 | fmptd 6385 | . 2 |
21 | ffn 6045 | . . . . . 6 | |
22 | 1, 21 | syl 17 | . . . . 5 |
23 | ffn 6045 | . . . . . 6 | |
24 | 8, 23 | syl 17 | . . . . 5 |
25 | off2.4 | . . . . 5 | |
26 | off2.5 | . . . . 5 | |
27 | eqid 2622 | . . . . 5 | |
28 | eqidd 2623 | . . . . 5 | |
29 | eqidd 2623 | . . . . 5 | |
30 | 22, 24, 25, 26, 27, 28, 29 | offval 6904 | . . . 4 |
31 | 3 | mpteq1d 4738 | . . . 4 |
32 | 30, 31 | eqtrd 2656 | . . 3 |
33 | 32 | feq1d 6030 | . 2 |
34 | 20, 33 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cin 3573 cmpt 4729 wfn 5883 wf 5884 cfv 5888 (class class class)co 6650 cof 6895 |
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-rep 4771 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-iun 4522 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-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 |
This theorem is referenced by: (None) |
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