Theorem rexrn 7059

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 6873	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V)
2		fvelrnb 6921	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥))
3		eqcom 2736	. . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
4	3	rexbii 3076	. . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
5	2, 4	bitrdi 287	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)))
6		rexrn.1	. . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
7	6	adantl 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓))
8	1, 5, 7	rexxfr2d 5366	1 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447  ran crn 5639   Fn wfn 6506  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  elrnrexdm  7061  wemapwe  9650  rexanuz  15312  climsup  15636  supcvg  15822  ruclem12  16209  prmreclem6  16892  vdwmc  16949  znunit  21473  lmbr2  23146  lmff  23188  1stcfb  23332  imasf1oxms  24377  lebnumlem3  24862  lmmbr2  25159  lmcau  25213  bcthlem4  25227  mbfsup  25565  itg2monolem1  25651  itg2gt0  25661  ostth  27550  uhgrvtxedgiedgb  29063  dfnbgr3  29265  vdn0conngrumgrv2  30125  erdszelem10  35187  neibastop2lem  36348  filnetlem4  36369  mblfinlem2  37652  istotbnd3  37765  sstotbnd  37769  heibor  37815  nacsfix  42700  fnwe2lem2  43040  climinf  45604  dfclnbgr3  47824