Theorem bren 8520

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 8519	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2		f1ofn 6618	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴)
3		fndm 6457	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴)
4		vex 3499	. . . . . . 7 ⊢ 𝑓 ∈ V
5	4	dmex 7618	. . . . . 6 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2924	. . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V)
7	2, 6	syl 17	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
8		f1ofo 6624	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
9		forn 6595	. . . . . 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 7619	. . . . 5 ⊢ ran 𝑓 ∈ V
12	10, 11	eqeltrrdi 2924	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
13	7, 12	jca 514	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	13	exlimiv 1931	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
15		f1oeq2 6607	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦))
16	15	exbidv 1922	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦))
17		f1oeq3 6608	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵))
18	17	exbidv 1922	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
19		df-en 8512	. . 3 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
20	16, 18, 19	brabg 5428	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
21	1, 14, 20	pm5.21nii 382	1 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3496   class class class wbr 5068  dom cdm 5557  ran crn 5558   Fn wfn 6352  –onto→wfo 6355  –1-1-onto→wf1o 6356   ≈ cen 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-en 8512

This theorem is referenced by:  domen  8524  f1oen3g  8527  ener  8558  en0  8574  ensn1  8575  en1  8578  unen  8598  enfixsn  8628  canth2  8672  mapen  8683  ssenen  8693  phplem4  8701  php3  8705  isinf  8733  ssfi  8740  domunfican  8793  fiint  8797  mapfien2  8874  unxpwdom2  9054  isinffi  9423  infxpenc2  9450  fseqen  9455  dfac8b  9459  infpwfien  9490  dfac12r  9574  infmap2  9642  cff1  9682  infpssr  9732  fin4en1  9733  enfin2i  9745  enfin1ai  9808  axcc3  9862  axcclem  9881  numth  9896  ttukey2g  9940  canthnum  10073  canthwe  10075  canthp1  10078  pwfseq  10088  tskuni  10207  gruen  10236  hasheqf1o  13712  hashfacen  13815  fz1f1o  15069  ruc  15598  cnso  15602  eulerth  16122  ablfaclem3  19211  lbslcic  20987  uvcendim  20993  indishmph  22408  ufldom  22572  ovolctb  24093  ovoliunlem3  24107  iunmbl2  24160  dyadmbl  24203  vitali  24216  cusgrfilem3  27241  padct  30457  f1ocnt  30527  volmeas  31492  eulerpart  31642  derangenlem  32420  mblfinlem1  34931  eldioph2lem1  39364  isnumbasgrplem1  39708  nnf1oxpnn  41464  sprsymrelen  43669  prproropen  43677  uspgrspren  44034  uspgrbisymrel  44036  rrx2xpreen  44713