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Theorem ralima 7256
Description: Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025.)
Hypothesis
Ref Expression
ralima.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
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem ralima
StepHypRef Expression
1 fnfun 6668 . . 3 (𝐹 Fn 𝐴 → Fun 𝐹)
21funfnd 6598 . 2 (𝐹 Fn 𝐴𝐹 Fn dom 𝐹)
3 fndm 6671 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43sseq2d 4027 . . 3 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
54biimpar 477 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
6 fvexd 6921 . . 3 (((𝐹 Fn dom 𝐹𝐵 ⊆ dom 𝐹) ∧ 𝑦𝐵) → (𝐹𝑦) ∈ V)
7 fvelimab 6980 . . . 4 ((𝐹 Fn dom 𝐹𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
8 eqcom 2741 . . . . 5 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
98rexbii 3091 . . . 4 (∃𝑦𝐵 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦))
107, 9bitrdi 287 . . 3 ((𝐹 Fn dom 𝐹𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
11 ralima.x . . . 4 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
1211adantl 481 . . 3 (((𝐹 Fn dom 𝐹𝐵 ⊆ dom 𝐹) ∧ 𝑥 = (𝐹𝑦)) → (𝜑𝜓))
136, 10, 12ralxfr2d 5415 . 2 ((𝐹 Fn dom 𝐹𝐵 ⊆ dom 𝐹) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))
142, 5, 13syl2an2r 685 1 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  wss 3962  dom cdm 5688  cima 5691   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  rexima  7257  supisolem  9510  ordtypelem6  9560  ordtypelem7  9561  limsupgle  15509  mrcuni  17665  ipodrsima  18598  mgmhmima  18740  mhmimalem  18849  ghmnsgima  19270  cntzmhm  19371  rhmimasubrnglem  20581  qtopeu  23739  kqdisj  23755  ghmcnp  24138  qustgplem  24144  qtopbaslem  24794  bndth  25003  fmcfil  25319  ovoliunlem1  25550  volsup2  25653  mbflimsup  25714  itg2gt0  25809  mdegleb  26117  efopn  26714  fsumdvdsmul  27252  fsumdvdsmulOLD  27254  negsunif  28101  negsbdaylem  28102  imaelshi  32086  cvmopnlem  35262  weiunfrlem  36446  ovoliunnfl  37648  voliunnfl  37650  volsupnfl  37651  gicabl  43087
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