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Theorem ralima 7257
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 6597 . 2 (𝐹 Fn 𝐴𝐹 Fn dom 𝐹)
3 fndm 6671 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43sseq2d 4016 . . 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 6981 . . . 4 ((𝐹 Fn dom 𝐹𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
8 eqcom 2744 . . . . 5 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
98rexbii 3094 . . . 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 5410 . 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 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  dom cdm 5685  cima 5688   Fn wfn 6556  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  rexima  7258  supisolem  9513  ordtypelem6  9563  ordtypelem7  9564  limsupgle  15513  mrcuni  17664  ipodrsima  18586  mgmhmima  18728  mhmimalem  18837  ghmnsgima  19258  cntzmhm  19359  rhmimasubrnglem  20565  qtopeu  23724  kqdisj  23740  ghmcnp  24123  qustgplem  24129  qtopbaslem  24779  bndth  24990  fmcfil  25306  ovoliunlem1  25537  volsup2  25640  mbflimsup  25701  itg2gt0  25795  mdegleb  26103  efopn  26700  fsumdvdsmul  27238  fsumdvdsmulOLD  27240  negsunif  28087  negsbdaylem  28088  imaelshi  32077  cvmopnlem  35283  weiunfrlem  36465  ovoliunnfl  37669  voliunnfl  37671  volsupnfl  37672  gicabl  43111
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