Theorem rnmpt 5279

Description: The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rnmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
rnmpt	⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem rnmpt

Step	Hyp	Ref	Expression
1		rnopab 5278	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		rnmpt.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		df-mpt 4639	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	2, 3	eqtri 2631	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5	4	rneqi 5260	. 2 ⊢ ran 𝐹 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6		df-rex 2901	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
7	6	abbii 2725	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
8	1, 5, 7	3eqtr4i 2641	1 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Syntax hints:   ∧ wa 382   = wceq 1474  ∃wex 1694   ∈ wcel 1976  {cab 2595  ∃wrex 2896  {copab 4636   ↦ cmpt 4637  ran crn 5029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-cnv 5036  df-dm 5038  df-rn 5039

This theorem is referenced by:  elrnmpt  5280  elrnmpt1  5282  elrnmptg  5283  dfiun3g  5286  dfiin3g  5287  fnrnfv  6137  fmpt  6274  fnasrn  6302  fliftf  6443  abrexex  7010  abrexexg  7011  fo1st  7056  fo2nd  7057  qliftf  7699  abrexfi  8126  iinfi  8183  tz9.12lem1  8510  infmap2  8900  cfslb2n  8950  fin23lem29  9023  fin23lem30  9024  fin1a2lem11  9092  ac6num  9161  rankcf  9455  tskuni  9461  negfi  10820  4sqlem11  15443  4sqlem12  15444  vdwapval  15461  vdwlem6  15474  quslem  15972  conjnmzb  17464  pmtrprfvalrn  17677  sylow1lem2  17783  sylow3lem1  17811  sylow3lem2  17812  rnascl  19110  ellspd  19902  iinopn  20474  restco  20720  pnrmopn  20899  cncmp  20947  discmp  20953  comppfsc  21087  alexsublem  21600  ptcmplem3  21610  snclseqg  21671  prdsxmetlem  21924  prdsbl  22047  xrhmeo  22484  pi1xfrf  22592  pi1cof  22598  iunmbl  23045  voliun  23046  itg1addlem4  23189  i1fmulc  23193  mbfi1fseqlem4  23208  itg2monolem1  23240  aannenlem2  23805  2lgslem1b  24834  mptelee  25493  nbgraf1olem5  25740  fargshiftfo  25932  disjrnmpt  28586  ofrn2  28628  abrexct  28688  abrexctf  28690  esumc  29246  esumrnmpt  29247  carsgclctunlem3  29515  eulerpartlemt  29566  bdayfo  30880  fobigcup  30983  ptrest  32381  areacirclem2  32474  istotbnd3  32543  sstotbnd  32547  rmxypairf1o  36297  hbtlem6  36521  elrnmptf  38165  fnrnafv  39696