Theorem ralima 7233

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 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralima	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		fnfun 6633	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	funfnd 6564	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹)
3		fndm 6636	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	sseq2d 3977	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
5	4	biimpar 482	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
6		fvexd 6894	. . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V)
7		fvelimab 6951	. . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
8		eqcom 2776	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
9	8	rexbii 3118	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
10	7, 9	bitrdi 290	. . 3 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
11		ralima.x	. . . 4 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
12	11	adantl 486	. . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓))
13	6, 10, 12	ralxfr2d 5379	. 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
14	2, 5, 13	syl2an2r 697	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  dom cdm 5659   “ cima 5662   Fn wfn 6528  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is referenced by:  rexima  7234  supisolem  9430  ordtypelem6  9481  ordtypelem7  9482  limsupgle  15524  mrcuni  17673  ipodrsima  18593  mgmhmima  18769  mhmimalem  18879  ghmnsgima  19306  cntzmhm  19407  rhmimasubrnglem  20646  qtopeu  23838  kqdisj  23854  ghmcnp  24237  qustgplem  24243  qtopbaslem  24880  bndth  25082  fmcfil  25396  ovoliunlem1  25626  volsup2  25729  mbflimsup  25790  itg2gt0  25884  mdegleb  26186  efopn  26785  fsumdvdsmul  27321  negsunif  28210  negbdaylem  28211  oniso  28426  bdayn0p1  28524  imaelshi  32347  vonf1wev  35487  vonf1owevOLD  35489  cvmopnlem  35665  weiunfrlem  36860  ovoliunnfl  38196  voliunnfl  38198  volsupnfl  38199  gicabl  43711  permac8prim  45608