Theorem bren 8173

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 8172	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2		f1ofn 6325	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴)
3		fndm 6170	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4		vex 3353	. . . . . . 7 ⊢ 𝑓 ∈ V
5	4	dmex 7301	. . . . . 6 ⊢ dom 𝑓 ∈ V
6	3, 5	syl6eqelr 2853	. . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V)
7	2, 6	syl 17	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
8		f1ofo 6331	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
9		forn 6303	. . . . . 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 7302	. . . . 5 ⊢ ran 𝑓 ∈ V
12	10, 11	syl6eqelr 2853	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
13	7, 12	jca 507	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	13	exlimiv 2025	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
15		f1oeq2 6315	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦))
16	15	exbidv 2016	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦))
17		f1oeq3 6316	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵))
18	17	exbidv 2016	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
19		df-en 8165	. . 3 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
20	16, 18, 19	brabg 5157	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
21	1, 14, 20	pm5.21nii 369	1 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)

Syntax hints:   ↔ wb 197   ∧ wa 384   = wceq 1652  ∃wex 1874   ∈ wcel 2155  Vcvv 3350   class class class wbr 4811  dom cdm 5279  ran crn 5280   Fn wfn 6065  –onto→wfo 6068  –1-1-onto→wf1o 6069   ≈ cen 8161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-xp 5285  df-rel 5286  df-cnv 5287  df-dm 5289  df-rn 5290  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-en 8165

This theorem is referenced by:  domen  8177  f1oen3g  8180  ener  8211  en0  8227  ensn1  8228  en1  8231  unen  8251  enfixsn  8280  canth2  8324  mapen  8335  ssenen  8345  phplem4  8353  php3  8357  isinf  8384  ssfi  8391  domunfican  8444  fiint  8448  mapfien2  8525  unxpwdom2  8704  isinffi  9073  infxpenc2  9100  fseqen  9105  dfac8b  9109  infpwfien  9140  dfac12r  9225  infmap2  9297  cff1  9337  infpssr  9387  fin4en1  9388  enfin2i  9400  enfin1ai  9463  axcc3  9517  axcclem  9536  numth  9551  ttukey2g  9595  canthnum  9728  canthwe  9730  canthp1  9733  pwfseq  9743  tskuni  9862  gruen  9891  hasheqf1o  13346  hashfacen  13444  fz1f1o  14740  ruc  15268  cnso  15272  eulerth  15781  ablfaclem3  18767  lbslcic  20470  uvcendim  20476  indishmph  21895  ufldom  22059  ovolctb  23562  ovoliunlem3  23576  iunmbl2  23629  dyadmbl  23672  vitali  23685  cusgrfilem3  26658  padct  29967  f1ocnt  30029  volmeas  30762  eulerpart  30912  derangenlem  31622  mblfinlem1  33891  eldioph2lem1  38025  isnumbasgrplem1  38372  nnf1oxpnn  40055  sprsymrelen  42443  uspgrspren  42453  uspgrbisymrel  42455