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Mirrors > Home > MPE Home > Th. List > fveqressseq | Structured version Visualization version Unicode version |
Description: If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.) |
Ref | Expression |
---|---|
fveqdmss.1 |
Ref | Expression |
---|---|
fveqressseq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqdmss.1 | . . . 4 | |
2 | 1 | fveqdmss 6354 | . . 3 |
3 | dmres 5419 | . . . . 5 | |
4 | incom 3805 | . . . . . 6 | |
5 | sseqin2 3817 | . . . . . . 7 | |
6 | 5 | biimpi 206 | . . . . . 6 |
7 | 4, 6 | syl5eq 2668 | . . . . 5 |
8 | 3, 7 | syl5eq 2668 | . . . 4 |
9 | 8, 1 | syl6eq 2672 | . . 3 |
10 | 2, 9 | syl 17 | . 2 |
11 | fvres 6207 | . . . . . . . 8 | |
12 | 11 | adantl 482 | . . . . . . 7 |
13 | id 22 | . . . . . . 7 | |
14 | 12, 13 | sylan9eq 2676 | . . . . . 6 |
15 | 14 | ex 450 | . . . . 5 |
16 | 15 | ralimdva 2962 | . . . 4 |
17 | 16 | 3impia 1261 | . . 3 |
18 | 2, 7 | syl 17 | . . . . 5 |
19 | 3, 18 | syl5eq 2668 | . . . 4 |
20 | 19 | raleqdv 3144 | . . 3 |
21 | 17, 20 | mpbird 247 | . 2 |
22 | simpll 790 | . . . . . . . 8 | |
23 | 1 | eleq2i 2693 | . . . . . . . . . 10 |
24 | 23 | biimpi 206 | . . . . . . . . 9 |
25 | 24 | adantl 482 | . . . . . . . 8 |
26 | simplr 792 | . . . . . . . 8 | |
27 | nelrnfvne 6353 | . . . . . . . 8 | |
28 | 22, 25, 26, 27 | syl3anc 1326 | . . . . . . 7 |
29 | neeq1 2856 | . . . . . . 7 | |
30 | 28, 29 | syl5ibrcom 237 | . . . . . 6 |
31 | 30 | ralimdva 2962 | . . . . 5 |
32 | 31 | 3impia 1261 | . . . 4 |
33 | fvn0ssdmfun 6350 | . . . . 5 | |
34 | 33 | simprd 479 | . . . 4 |
35 | 32, 34 | syl 17 | . . 3 |
36 | simp1 1061 | . . 3 | |
37 | eqfunfv 6316 | . . 3 | |
38 | 35, 36, 37 | syl2anc 693 | . 2 |
39 | 10, 21, 38 | mpbir2and 957 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wnel 2897 wral 2912 cin 3573 wss 3574 c0 3915 cdm 5114 crn 5115 cres 5116 wfun 5882 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-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 |
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-nel 2898 df-ral 2917 df-rex 2918 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-fv 5896 |
This theorem is referenced by: plusfreseq 41772 |
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