Theorem bren 6031

Description: Equinumerosity relationship. (Contributed by SF, 23-Feb-2015.)

Assertion

Ref	Expression
bren	⊢ (A ≈ B ↔ ∃f f:A–1-1-onto→B)

Distinct variable groups:   A,f   B,f

Proof of Theorem bren

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. 2 ⊢ (A ≈ B → (A ∈ V ∧ B ∈ V))
2		vex 2863	. . . . . 6 ⊢ f ∈ V
3	2	dmex 5107	. . . . 5 ⊢ dom f ∈ V
4	2	rnex 5108	. . . . 5 ⊢ ran f ∈ V
5	3, 4	pm3.2i 441	. . . 4 ⊢ (dom f ∈ V ∧ ran f ∈ V)
6		f1odm 5291	. . . . . 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 5294	. . . . . . 7 ⊢ (f:A–1-1-onto→B → f:A–onto→B)
9		forn 5273	. . . . . . 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 5283	. . . 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 5284	. . . 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 6030	. . 3 ⊢ ≈ = {⟨x, y⟩ ∣ ∃f f:x–1-1-onto→y}
20	16, 18, 19	brabg 4707	. 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)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   class class class wbr 4640  dom cdm 4773  ran crn 4774  –onto→wfo 4780  –1-1-onto→wf1o 4781   ≈ cen 6029

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-swap 4725  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-en 6030

This theorem is referenced by:  f1oeng  6033  ensymi  6037  entr  6039  en0  6043  unen  6049  xpen  6056  enpw1  6063  enmap2  6069  enmap1  6075  nenpw1pwlem2  6086  ncdisjun  6137  1cnc  6140  sbthlem3  6206  nclenc  6223  lenc  6224