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Mirrors > Home > NFE Home > Th. List > fnfullfunlem1 | Unicode version |
Description: Lemma for fnfullfun 5858. Binary relationship over part one of the full function definition. (Contributed by SF, 9-Mar-2015.) |
Ref | Expression |
---|---|
fnfullfunlem1 | ∼ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . . 3 ∼ | |
2 | 1 | simprd 449 | . 2 ∼ |
3 | brex 4689 | . . . 4 | |
4 | 3 | simprd 449 | . . 3 |
5 | 4 | adantr 451 | . 2 |
6 | breq2 4643 | . . . 4 ∼ ∼ | |
7 | breq2 4643 | . . . . 5 | |
8 | eqeq2 2362 | . . . . . . 7 | |
9 | 8 | imbi2d 307 | . . . . . 6 |
10 | 9 | albidv 1625 | . . . . 5 |
11 | 7, 10 | anbi12d 691 | . . . 4 |
12 | 6, 11 | bibi12d 312 | . . 3 ∼ ∼ |
13 | brdif 4694 | . . . 4 ∼ ∼ | |
14 | coi2 5095 | . . . . . 6 | |
15 | 14 | breqi 4645 | . . . . 5 |
16 | brco 4883 | . . . . . . 7 ∼ ∼ | |
17 | df-br 4640 | . . . . . . . . . 10 ∼ ∼ | |
18 | vex 2862 | . . . . . . . . . . . . 13 | |
19 | vex 2862 | . . . . . . . . . . . . 13 | |
20 | 18, 19 | opex 4588 | . . . . . . . . . . . 12 |
21 | 20 | elcompl 3225 | . . . . . . . . . . 11 ∼ |
22 | df-br 4640 | . . . . . . . . . . . 12 | |
23 | 19 | ideq 4870 | . . . . . . . . . . . 12 |
24 | 22, 23 | bitr3i 242 | . . . . . . . . . . 11 |
25 | 21, 24 | xchbinx 301 | . . . . . . . . . 10 ∼ |
26 | 17, 25 | bitri 240 | . . . . . . . . 9 ∼ |
27 | 26 | anbi2i 675 | . . . . . . . 8 ∼ |
28 | 27 | exbii 1582 | . . . . . . 7 ∼ |
29 | exanali 1585 | . . . . . . 7 | |
30 | 16, 28, 29 | 3bitrri 263 | . . . . . 6 ∼ |
31 | 30 | con1bii 321 | . . . . 5 ∼ |
32 | 15, 31 | anbi12i 678 | . . . 4 ∼ |
33 | 13, 32 | bitri 240 | . . 3 ∼ |
34 | 12, 33 | vtoclg 2914 | . 2 ∼ |
35 | 2, 5, 34 | pm5.21nii 342 | 1 ∼ |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wal 1540 wex 1541 wceq 1642 wcel 1710 cvv 2859 ∼ ccompl 3205 cdif 3206 cop 4561 class class class wbr 4639 ccom 4721 cid 4763 |
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-co 4726 df-id 4767 df-cnv 4785 |
This theorem is referenced by: fnfullfunlem2 5857 fvfullfunlem1 5861 fvfullfunlem2 5862 |
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