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Mirrors > Home > MPE Home > Th. List > fparlem3 | Structured version Visualization version Unicode version |
Description: Lemma for fpar 7281. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fparlem3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coiun 5645 | . 2 | |
2 | inss1 3833 | . . . . 5 | |
3 | fndm 5990 | . . . . 5 | |
4 | 2, 3 | syl5sseq 3653 | . . . 4 |
5 | dfco2a 5635 | . . . 4 | |
6 | 4, 5 | syl 17 | . . 3 |
7 | 6 | coeq2d 5284 | . 2 |
8 | inss1 3833 | . . . . . . . . 9 | |
9 | dmxpss 5565 | . . . . . . . . 9 | |
10 | 8, 9 | sstri 3612 | . . . . . . . 8 |
11 | dfco2a 5635 | . . . . . . . 8 | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 |
13 | fvex 6201 | . . . . . . . 8 | |
14 | fparlem1 7277 | . . . . . . . . . 10 | |
15 | sneq 4187 | . . . . . . . . . . 11 | |
16 | 15 | xpeq1d 5138 | . . . . . . . . . 10 |
17 | 14, 16 | syl5eq 2668 | . . . . . . . . 9 |
18 | 15 | imaeq2d 5466 | . . . . . . . . . 10 |
19 | df-ima 5127 | . . . . . . . . . . 11 | |
20 | ssid 3624 | . . . . . . . . . . . . . 14 | |
21 | xpssres 5434 | . . . . . . . . . . . . . 14 | |
22 | 20, 21 | ax-mp 5 | . . . . . . . . . . . . 13 |
23 | 22 | rneqi 5352 | . . . . . . . . . . . 12 |
24 | 13 | snnz 4309 | . . . . . . . . . . . . 13 |
25 | rnxp 5564 | . . . . . . . . . . . . 13 | |
26 | 24, 25 | ax-mp 5 | . . . . . . . . . . . 12 |
27 | 23, 26 | eqtri 2644 | . . . . . . . . . . 11 |
28 | 19, 27 | eqtri 2644 | . . . . . . . . . 10 |
29 | 18, 28 | syl6eq 2672 | . . . . . . . . 9 |
30 | 17, 29 | xpeq12d 5140 | . . . . . . . 8 |
31 | 13, 30 | iunxsn 4603 | . . . . . . 7 |
32 | 12, 31 | eqtri 2644 | . . . . . 6 |
33 | 32 | cnveqi 5297 | . . . . 5 |
34 | cnvco 5308 | . . . . 5 | |
35 | cnvxp 5551 | . . . . 5 | |
36 | 33, 34, 35 | 3eqtr3i 2652 | . . . 4 |
37 | fparlem1 7277 | . . . . . . . . 9 | |
38 | 37 | xpeq2i 5136 | . . . . . . . 8 |
39 | fnsnfv 6258 | . . . . . . . . 9 | |
40 | 39 | xpeq1d 5138 | . . . . . . . 8 |
41 | 38, 40 | syl5eqr 2670 | . . . . . . 7 |
42 | 41 | cnveqd 5298 | . . . . . 6 |
43 | cnvxp 5551 | . . . . . 6 | |
44 | 42, 43 | syl6eq 2672 | . . . . 5 |
45 | 44 | coeq2d 5284 | . . . 4 |
46 | 36, 45 | syl5eqr 2670 | . . 3 |
47 | 46 | iuneq2dv 4542 | . 2 |
48 | 1, 7, 47 | 3eqtr4a 2682 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 cin 3573 wss 3574 c0 3915 csn 4177 ciun 4520 cxp 5112 ccnv 5113 cdm 5114 crn 5115 cres 5116 cima 5117 ccom 5118 wfn 5883 cfv 5888 c1st 7166 |
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-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-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: fpar 7281 |
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