Theorem rexrn 7035

Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Hypothesis

Ref	Expression
rexrn.1	⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexrn	⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rexrn

Step	Hyp	Ref	Expression
1		fvexd 6849	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V)
2		fvelrnb 6894	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥))
3		eqcom 2747	. . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
4	3	rexbii 3087	. . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
5	2, 4	bitrdi 288	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)))
6		rexrn.1	. . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
7	6	adantl 482	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓))
8	1, 5, 7	rexxfr2d 5347	1 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  Vcvv 3432  ran crn 5626   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  elrnrexdm  7037  wemapwe  9616  rexanuz  15306  climsup  15630  supcvg  15819  ruclem12  16206  prmreclem6  16890  vdwmc  16947  znunit  21545  lmbr2  23249  lmff  23291  1stcfb  23435  imasf1oxms  24479  lebnumlem3  24955  lmmbr2  25251  lmcau  25305  bcthlem4  25319  mbfsup  25656  itg2monolem1  25742  itg2gt0  25752  ostth  27627  uhgrvtxedgiedgb  29230  dfnbgr3  29432  vdn0conngrumgrv2  30291  erdszelem10  35435  neibastop2lem  36595  filnetlem4  36616  mblfinlem2  38032  istotbnd3  38145  sstotbnd  38149  heibor  38195  nacsfix  43168  fnwe2lem2  43503  climinf  46058  dfclnbgr3  48324