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Mirrors > Home > NFE Home > Th. List > enex | Unicode version |
Description: The equinumerosity relationship is a set. (Contributed by SF, 23-Feb-2015.) |
Ref | Expression |
---|---|
enex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-en 6029 | . . 3 | |
2 | elrn2 4897 | . . . . 5 Fns Image Swap Fns Fns Image Swap Fns | |
3 | df-br 4640 | . . . . . . . . 9 Fns Fns | |
4 | vex 2862 | . . . . . . . . . 10 | |
5 | 4 | brfns 5833 | . . . . . . . . 9 Fns |
6 | 3, 5 | bitr3i 242 | . . . . . . . 8 Fns |
7 | elrn2 4897 | . . . . . . . . 9 Image Swap Fns Image Swap Fns | |
8 | oteltxp 5782 | . . . . . . . . . . . 12 Image Swap Fns Image Swap Fns | |
9 | opelcnv 4893 | . . . . . . . . . . . . . 14 Image Swap Image Swap | |
10 | dfcnv2 5100 | . . . . . . . . . . . . . . . 16 Swap | |
11 | 10 | eqeq2i 2363 | . . . . . . . . . . . . . . 15 Swap |
12 | vex 2862 | . . . . . . . . . . . . . . . 16 | |
13 | 4, 12 | brimage 5793 | . . . . . . . . . . . . . . 15 Image Swap Swap |
14 | df-br 4640 | . . . . . . . . . . . . . . 15 Image Swap Image Swap | |
15 | 11, 13, 14 | 3bitr2ri 265 | . . . . . . . . . . . . . 14 Image Swap |
16 | 9, 15 | bitri 240 | . . . . . . . . . . . . 13 Image Swap |
17 | df-br 4640 | . . . . . . . . . . . . . 14 Fns Fns | |
18 | 12 | brfns 5833 | . . . . . . . . . . . . . 14 Fns |
19 | 17, 18 | bitr3i 242 | . . . . . . . . . . . . 13 Fns |
20 | 16, 19 | anbi12i 678 | . . . . . . . . . . . 12 Image Swap Fns |
21 | 8, 20 | bitri 240 | . . . . . . . . . . 11 Image Swap Fns |
22 | 21 | exbii 1582 | . . . . . . . . . 10 Image Swap Fns |
23 | 4 | cnvex 5102 | . . . . . . . . . . 11 |
24 | fneq1 5173 | . . . . . . . . . . 11 | |
25 | 23, 24 | ceqsexv 2894 | . . . . . . . . . 10 |
26 | 22, 25 | bitri 240 | . . . . . . . . 9 Image Swap Fns |
27 | 7, 26 | bitri 240 | . . . . . . . 8 Image Swap Fns |
28 | 6, 27 | anbi12i 678 | . . . . . . 7 Fns Image Swap Fns |
29 | oteltxp 5782 | . . . . . . 7 Fns Image Swap Fns Fns Image Swap Fns | |
30 | dff1o4 5294 | . . . . . . 7 | |
31 | 28, 29, 30 | 3bitr4i 268 | . . . . . 6 Fns Image Swap Fns |
32 | 31 | exbii 1582 | . . . . 5 Fns Image Swap Fns |
33 | 2, 32 | bitri 240 | . . . 4 Fns Image Swap Fns |
34 | 33 | opabbi2i 4866 | . . 3 Fns Image Swap Fns |
35 | 1, 34 | eqtr4i 2376 | . 2 Fns Image Swap Fns |
36 | fnsex 5832 | . . . 4 Fns | |
37 | swapex 4742 | . . . . . . . 8 Swap | |
38 | 37 | imageex 5801 | . . . . . . 7 Image Swap |
39 | 38 | cnvex 5102 | . . . . . 6 Image Swap |
40 | 39, 36 | txpex 5785 | . . . . 5 Image Swap Fns |
41 | 40 | rnex 5107 | . . . 4 Image Swap Fns |
42 | 36, 41 | txpex 5785 | . . 3 Fns Image Swap Fns |
43 | 42 | rnex 5107 | . 2 Fns Image Swap Fns |
44 | 35, 43 | eqeltri 2423 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 358 wex 1541 wceq 1642 wcel 1710 cvv 2859 cop 4561 copab 4622 class class class wbr 4639 Swap cswap 4718 cima 4722 ccnv 4771 crn 4773 wfn 4776 wf1o 4780 ctxp 5735 Imagecimage 5753 Fns cfns 5761 cen 6028 |
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-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 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 df-2nd 4797 df-txp 5736 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-en 6029 |
This theorem is referenced by: ener 6039 ncsex 6111 ncex 6117 ovmuc 6130 mucex 6133 ovcelem1 6171 ceex 6174 |
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