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Mirrors > Home > NFE Home > Th. List > txpcofun | Unicode version |
Description: Composition distributes over tail cross product in the case of a function. (Contributed by SF, 18-Feb-2015.) |
Ref | Expression |
---|---|
txpcofun.1 |
Ref | Expression |
---|---|
txpcofun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . 4 | |
2 | opeqex 4621 | . . . 4 | |
3 | 1, 2 | ax-mp 8 | . . 3 |
4 | dmcoss 4971 | . . . . . . . . . 10 | |
5 | opeldm 4910 | . . . . . . . . . 10 | |
6 | 4, 5 | sseldi 3271 | . . . . . . . . 9 |
7 | 6 | pm4.71ri 614 | . . . . . . . 8 |
8 | 7 | anbi1i 676 | . . . . . . 7 |
9 | anass 630 | . . . . . . 7 | |
10 | fvex 5339 | . . . . . . . . . . 11 | |
11 | breq1 4642 | . . . . . . . . . . 11 | |
12 | 10, 11 | ceqsexv 2894 | . . . . . . . . . 10 |
13 | breq1 4642 | . . . . . . . . . . 11 | |
14 | 10, 13 | ceqsexv 2894 | . . . . . . . . . 10 |
15 | 12, 14 | anbi12i 678 | . . . . . . . . 9 |
16 | eqcom 2355 | . . . . . . . . . . . . . 14 | |
17 | txpcofun.1 | . . . . . . . . . . . . . . 15 | |
18 | funbrfvb 5360 | . . . . . . . . . . . . . . 15 | |
19 | 17, 18 | mpan 651 | . . . . . . . . . . . . . 14 |
20 | 16, 19 | syl5bb 248 | . . . . . . . . . . . . 13 |
21 | 20 | anbi1d 685 | . . . . . . . . . . . 12 |
22 | 21 | exbidv 1626 | . . . . . . . . . . 11 |
23 | opelco 4884 | . . . . . . . . . . 11 | |
24 | 22, 23 | syl6bbr 254 | . . . . . . . . . 10 |
25 | 20 | anbi1d 685 | . . . . . . . . . . . 12 |
26 | 25 | exbidv 1626 | . . . . . . . . . . 11 |
27 | opelco 4884 | . . . . . . . . . . 11 | |
28 | 26, 27 | syl6bbr 254 | . . . . . . . . . 10 |
29 | 24, 28 | anbi12d 691 | . . . . . . . . 9 |
30 | 15, 29 | syl5rbbr 251 | . . . . . . . 8 |
31 | 30 | pm5.32i 618 | . . . . . . 7 |
32 | 8, 9, 31 | 3bitrri 263 | . . . . . 6 |
33 | opelco 4884 | . . . . . . 7 | |
34 | 19.41v 1901 | . . . . . . . 8 | |
35 | funbrfv 5356 | . . . . . . . . . . . 12 | |
36 | 17, 35 | ax-mp 8 | . . . . . . . . . . 11 |
37 | trtxp 5781 | . . . . . . . . . . . 12 | |
38 | breq1 4642 | . . . . . . . . . . . 12 | |
39 | 37, 38 | syl5rbbr 251 | . . . . . . . . . . 11 |
40 | 36, 39 | syl 15 | . . . . . . . . . 10 |
41 | 40 | pm5.32i 618 | . . . . . . . . 9 |
42 | 41 | exbii 1582 | . . . . . . . 8 |
43 | eldm 4898 | . . . . . . . . 9 | |
44 | 43 | anbi1i 676 | . . . . . . . 8 |
45 | 34, 42, 44 | 3bitr4i 268 | . . . . . . 7 |
46 | 33, 45 | bitri 240 | . . . . . 6 |
47 | oteltxp 5782 | . . . . . 6 | |
48 | 32, 46, 47 | 3bitr4i 268 | . . . . 5 |
49 | opeq2 4579 | . . . . . . 7 | |
50 | 49 | eleq1d 2419 | . . . . . 6 |
51 | 49 | eleq1d 2419 | . . . . . 6 |
52 | 50, 51 | bibi12d 312 | . . . . 5 |
53 | 48, 52 | mpbiri 224 | . . . 4 |
54 | 53 | exlimivv 1635 | . . 3 |
55 | 3, 54 | ax-mp 8 | . 2 |
56 | 55 | eqrelriv 4850 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 cvv 2859 cop 4561 class class class wbr 4639 ccom 4721 cdm 4772 wfun 4775 cfv 4781 ctxp 5735 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fv 4795 df-2nd 4797 df-txp 5736 |
This theorem is referenced by: (None) |
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