Theorem bren 7850

Description: Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)

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		encv 7849	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2		f1ofn 6051	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴)
3		fndm 5904	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4		vex 3176	. . . . . . 7 ⊢ 𝑓 ∈ V
5	4	dmex 6991	. . . . . 6 ⊢ dom 𝑓 ∈ V
6	3, 5	syl6eqelr 2697	. . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V)
7	2, 6	syl 17	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
8		f1ofo 6057	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
9		forn 6031	. . . . . 6 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵)
10	8, 9	syl 17	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ran 𝑓 = 𝐵)
11	4	rnex 6992	. . . . 5 ⊢ ran 𝑓 ∈ V
12	10, 11	syl6eqelr 2697	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
13	7, 12	jca 553	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	13	exlimiv 1845	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
15		f1oeq2 6041	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦))
16	15	exbidv 1837	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦))
17		f1oeq3 6042	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵))
18	17	exbidv 1837	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
19		df-en 7842	. . 3 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
20	16, 18, 19	brabg 4919	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
21	1, 14, 20	pm5.21nii 367	1 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  dom cdm 5038  ran crn 5039   Fn wfn 5799  –onto→wfo 5802  –1-1-onto→wf1o 5803   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-en 7842

This theorem is referenced by:  domen  7854  f1oen3g  7857  ener  7888  enerOLD  7889  en0  7905  ensn1  7906  en1  7909  unen  7925  enfixsn  7954  canth2  7998  mapen  8009  ssenen  8019  phplem4  8027  php3  8031  isinf  8058  ssfi  8065  domunfican  8118  fiint  8122  mapfien2  8197  unxpwdom2  8376  isinffi  8701  infxpenc2  8728  fseqen  8733  dfac8b  8737  infpwfien  8768  dfac12r  8851  infmap2  8923  cff1  8963  infpssr  9013  fin4en1  9014  enfin2i  9026  enfin1ai  9089  axcc3  9143  axcclem  9162  numth  9177  ttukey2g  9221  canthnum  9350  canthwe  9352  canthp1  9355  pwfseq  9365  tskuni  9484  gruen  9513  hasheqf1o  12999  hashfacen  13095  fz1f1o  14288  ruc  14811  cnso  14815  eulerth  15326  ablfaclem3  18309  lbslcic  19999  uvcendim  20005  indishmph  21411  ufldom  21576  ovolctb  23065  ovoliunlem3  23079  iunmbl2  23132  dyadmbl  23174  vitali  23188  nbusgrafi  25977  cusgrafilem3  26009  wlknwwlknen  26243  padct  28885  f1ocnt  28946  volmeas  29621  eulerpart  29771  derangenlem  30407  mblfinlem1  32616  eldioph2lem1  36341  isnumbasgrplem1  36690  nnf1oxpnn  38379  cusgrfilem3  40673  wlknwwlksnen  41090