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Mirrors > Home > NFE Home > Th. List > swapf1o | Unicode version |
Description: Swap is a bijection over the universe. (Contributed by SF, 23-Feb-2015.) (Revised by Scott Fenton, 17-Apr-2021.) |
Ref | Expression |
---|---|
swapf1o | Swap |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun2 5119 | . . . 4 Swap Swap Swap | |
2 | opeq 4619 | . . . . . . . 8 Proj1 Proj2 | |
3 | 2 | breq2i 4647 | . . . . . . 7 Swap Swap Proj1 Proj2 |
4 | vex 2862 | . . . . . . . . 9 | |
5 | 4 | proj1ex 4593 | . . . . . . . 8 Proj1 |
6 | 4 | proj2ex 4594 | . . . . . . . 8 Proj2 |
7 | 5, 6 | brswap2 4860 | . . . . . . 7 Swap Proj1 Proj2 Proj2 Proj1 |
8 | 3, 7 | bitri 240 | . . . . . 6 Swap Proj2 Proj1 |
9 | opeq 4619 | . . . . . . . 8 Proj1 Proj2 | |
10 | 9 | breq2i 4647 | . . . . . . 7 Swap Swap Proj1 Proj2 |
11 | vex 2862 | . . . . . . . . 9 | |
12 | 11 | proj1ex 4593 | . . . . . . . 8 Proj1 |
13 | 11 | proj2ex 4594 | . . . . . . . 8 Proj2 |
14 | 12, 13 | brswap2 4860 | . . . . . . 7 Swap Proj1 Proj2 Proj2 Proj1 |
15 | 10, 14 | bitri 240 | . . . . . 6 Swap Proj2 Proj1 |
16 | eqtr2 2371 | . . . . . . 7 Proj2 Proj1 Proj2 Proj1 Proj2 Proj1 Proj2 Proj1 | |
17 | ancom 437 | . . . . . . . 8 Proj2 Proj2 Proj1 Proj1 Proj1 Proj1 Proj2 Proj2 | |
18 | opth 4602 | . . . . . . . 8 Proj2 Proj1 Proj2 Proj1 Proj2 Proj2 Proj1 Proj1 | |
19 | 2, 9 | eqeq12i 2366 | . . . . . . . . 9 Proj1 Proj2 Proj1 Proj2 |
20 | opth 4602 | . . . . . . . . 9 Proj1 Proj2 Proj1 Proj2 Proj1 Proj1 Proj2 Proj2 | |
21 | 19, 20 | bitri 240 | . . . . . . . 8 Proj1 Proj1 Proj2 Proj2 |
22 | 17, 18, 21 | 3bitr4i 268 | . . . . . . 7 Proj2 Proj1 Proj2 Proj1 |
23 | 16, 22 | sylib 188 | . . . . . 6 Proj2 Proj1 Proj2 Proj1 |
24 | 8, 15, 23 | syl2anb 465 | . . . . 5 Swap Swap |
25 | 24 | gen2 1547 | . . . 4 Swap Swap |
26 | 1, 25 | mpgbir 1550 | . . 3 Swap |
27 | eqv 3565 | . . . 4 Swap Swap | |
28 | opeq 4619 | . . . . 5 Proj1 Proj2 | |
29 | eqid 2353 | . . . . . . 7 Proj1 Proj2 Proj1 Proj2 | |
30 | vex 2862 | . . . . . . . . 9 | |
31 | 30 | proj2ex 4594 | . . . . . . . 8 Proj2 |
32 | 30 | proj1ex 4593 | . . . . . . . 8 Proj1 |
33 | 31, 32 | brswap2 4860 | . . . . . . 7 Proj1 Proj2 Swap Proj2 Proj1 Proj1 Proj2 Proj1 Proj2 |
34 | 29, 33 | mpbir 200 | . . . . . 6 Proj1 Proj2 Swap Proj2 Proj1 |
35 | breldm 4911 | . . . . . 6 Proj1 Proj2 Swap Proj2 Proj1 Proj1 Proj2 Swap | |
36 | 34, 35 | ax-mp 5 | . . . . 5 Proj1 Proj2 Swap |
37 | 28, 36 | eqeltri 2423 | . . . 4 Swap |
38 | 27, 37 | mpgbir 1550 | . . 3 Swap |
39 | df-fn 4790 | . . 3 Swap Swap Swap | |
40 | 26, 38, 39 | mpbir2an 886 | . 2 Swap |
41 | cnvswap 5510 | . . . 4 Swap Swap | |
42 | 41 | fneq1i 5178 | . . 3 Swap Swap |
43 | 40, 42 | mpbir 200 | . 2 Swap |
44 | dff1o4 5294 | . 2 Swap Swap Swap | |
45 | 40, 43, 44 | mpbir2an 886 | 1 Swap |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wal 1540 wceq 1642 wcel 1710 cvv 2859 cop 4561 Proj1 cproj1 4563 Proj2 cproj2 4564 class class class wbr 4639 Swap cswap 4718 ccnv 4771 cdm 4772 wfun 4775 wfn 4776 wf1o 4780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-swap 4724 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-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 |
This theorem is referenced by: swapres 5512 |
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