Theorem ralima 6498

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 = ( F ` y ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralima	|- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )

Distinct variable groups:   ph, y   ps, x   x, F, y   x, B, y   x, A, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem ralima

Step	Hyp	Ref	Expression
1		fvexd 6203	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ y e. B ) -> ( F ` y ) e. _V )
2		fvelimab 6253	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B ( F ` y ) = x ) )
3		eqcom 2629	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
4	3	rexbii 3041	. . 3 |- ( E. y e. B ( F ` y ) = x <-> E. y e. B x = ( F ` y ) )
5	2, 4	syl6bb 276	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( x e. ( F " B ) <-> E. y e. B x = ( F ` y ) ) )
6		rexima.x	. . 3 |- ( x = ( F ` y ) -> ( ph <-> ps ) )
7	6	adantl 482	. 2 |- ( ( ( F Fn A /\ B C_ A ) /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
8	1, 5, 7	ralxfr2d 4882	1 |- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   "cima 5117   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  supisolem  8379  ordtypelem6  8428  ordtypelem7  8429  limsupgle  14208  mrcuni  16281  ipodrsima  17165  mhmima  17363  ghmnsgima  17684  cntzmhm  17771  qtopeu  21519  kqdisj  21535  ghmcnp  21918  qustgplem  21924  qtopbaslem  22562  bndth  22757  fmcfil  23070  ovoliunlem1  23270  volsup2  23373  mbflimsup  23433  itg2gt0  23527  mdegleb  23824  efopn  24404  fsumdvdsmul  24921  imaelshi  28917  cvmopnlem  31260  ovoliunnfl  33451  voliunnfl  33453  volsupnfl  33454  gicabl  37669  mgmhmima  41802