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Mirrors > Home > NFE Home > Th. List > bren | GIF version |
Description: Equinumerosity relationship. (Contributed by SF, 23-Feb-2015.) |
Ref | Expression |
---|---|
bren | ⊢ (A ≈ B ↔ ∃f f:A–1-1-onto→B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . 2 ⊢ (A ≈ B → (A ∈ V ∧ B ∈ V)) | |
2 | vex 2862 | . . . . . 6 ⊢ f ∈ V | |
3 | 2 | dmex 5106 | . . . . 5 ⊢ dom f ∈ V |
4 | 2 | rnex 5107 | . . . . 5 ⊢ ran f ∈ V |
5 | 3, 4 | pm3.2i 441 | . . . 4 ⊢ (dom f ∈ V ∧ ran f ∈ V) |
6 | f1odm 5290 | . . . . . 6 ⊢ (f:A–1-1-onto→B → dom f = A) | |
7 | 6 | eleq1d 2419 | . . . . 5 ⊢ (f:A–1-1-onto→B → (dom f ∈ V ↔ A ∈ V)) |
8 | f1ofo 5293 | . . . . . . 7 ⊢ (f:A–1-1-onto→B → f:A–onto→B) | |
9 | forn 5272 | . . . . . . 7 ⊢ (f:A–onto→B → ran f = B) | |
10 | 8, 9 | syl 15 | . . . . . 6 ⊢ (f:A–1-1-onto→B → ran f = B) |
11 | 10 | eleq1d 2419 | . . . . 5 ⊢ (f:A–1-1-onto→B → (ran f ∈ V ↔ B ∈ V)) |
12 | 7, 11 | anbi12d 691 | . . . 4 ⊢ (f:A–1-1-onto→B → ((dom f ∈ V ∧ ran f ∈ V) ↔ (A ∈ V ∧ B ∈ V))) |
13 | 5, 12 | mpbii 202 | . . 3 ⊢ (f:A–1-1-onto→B → (A ∈ V ∧ B ∈ V)) |
14 | 13 | exlimiv 1634 | . 2 ⊢ (∃f f:A–1-1-onto→B → (A ∈ V ∧ B ∈ V)) |
15 | f1oeq2 5282 | . . . 4 ⊢ (x = A → (f:x–1-1-onto→y ↔ f:A–1-1-onto→y)) | |
16 | 15 | exbidv 1626 | . . 3 ⊢ (x = A → (∃f f:x–1-1-onto→y ↔ ∃f f:A–1-1-onto→y)) |
17 | f1oeq3 5283 | . . . 4 ⊢ (y = B → (f:A–1-1-onto→y ↔ f:A–1-1-onto→B)) | |
18 | 17 | exbidv 1626 | . . 3 ⊢ (y = B → (∃f f:A–1-1-onto→y ↔ ∃f f:A–1-1-onto→B)) |
19 | df-en 6029 | . . 3 ⊢ ≈ = {〈x, y〉 ∣ ∃f f:x–1-1-onto→y} | |
20 | 16, 18, 19 | brabg 4706 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ≈ B ↔ ∃f f:A–1-1-onto→B)) |
21 | 1, 14, 20 | pm5.21nii 342 | 1 ⊢ (A ≈ B ↔ ∃f f:A–1-1-onto→B) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 class class class wbr 4639 dom cdm 4772 ran crn 4773 –onto→wfo 4779 –1-1-onto→wf1o 4780 ≈ 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-swap 4724 df-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-en 6029 |
This theorem is referenced by: f1oeng 6032 ensymi 6036 entr 6038 en0 6042 unen 6048 xpen 6055 enpw1 6062 enmap2 6068 enmap1 6074 nenpw1pwlem2 6085 ncdisjun 6136 1cnc 6139 sbthlem3 6205 nclenc 6222 lenc 6223 |
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