MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralima Structured version   Visualization version   GIF version

Theorem ralima 6377
Description: Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypothesis
Ref Expression
rexima.x (𝑥 = (𝐹𝑦) → (𝜑𝜓))
Assertion
Ref Expression
ralima ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝐹,𝑦   𝑥,𝐵,𝑦   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ralima
StepHypRef Expression
1 fvex 6095 . . 3 (𝐹𝑦) ∈ V
21a1i 11 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑦𝐵) → (𝐹𝑦) ∈ V)
3 fvelimab 6145 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
4 eqcom 2613 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
54rexbii 3019 . . 3 (∃𝑦𝐵 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦))
63, 5syl6bb 274 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
7 rexima.x . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
87adantl 480 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹𝑦)) → (𝜑𝜓))
92, 6, 8ralxfr2d 4800 1 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  wrex 2893  Vcvv 3169  wss 3536  cima 5028   Fn wfn 5782  cfv 5787
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 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-fv 5795
This theorem is referenced by:  supisolem  8236  ordtypelem6  8285  ordtypelem7  8286  limsupgle  13999  mrcuni  16047  ipodrsima  16931  mhmima  17129  ghmnsgima  17450  cntzmhm  17537  qtopeu  21268  kqdisj  21284  ghmcnp  21667  qustgplem  21673  qtopbaslem  22301  bndth  22493  fmcfil  22793  ovoliunlem1  22991  volsup2  23093  mbflimsup  23153  itg2gt0  23247  mdegleb  23542  efopn  24118  fsumdvdsmul  24635  imaelshi  28104  cvmopnlem  30317  ovoliunnfl  32421  voliunnfl  32423  volsupnfl  32424  gicabl  36487  mgmhmima  41591
  Copyright terms: Public domain W3C validator