Theorem ralima 7211

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 6618	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	funfnd 6547	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹)
3		fndm 6621	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	sseq2d 3979	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
5	4	biimpar 477	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
6		fvexd 6873	. . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V)
7		fvelimab 6933	. . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
8		eqcom 2736	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
9	8	rexbii 3076	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
10	7, 9	bitrdi 287	. . 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 481	. . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓))
13	6, 10, 12	ralxfr2d 5365	. 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
14	2, 5, 13	syl2an2r 685	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  dom cdm 5638   “ cima 5641   Fn wfn 6506  ‘cfv 6511

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  rexima  7212  supisolem  9425  ordtypelem6  9476  ordtypelem7  9477  limsupgle  15443  mrcuni  17582  ipodrsima  18500  mgmhmima  18642  mhmimalem  18751  ghmnsgima  19172  cntzmhm  19273  rhmimasubrnglem  20474  qtopeu  23603  kqdisj  23619  ghmcnp  24002  qustgplem  24008  qtopbaslem  24646  bndth  24857  fmcfil  25172  ovoliunlem1  25403  volsup2  25506  mbflimsup  25567  itg2gt0  25661  mdegleb  25969  efopn  26567  fsumdvdsmul  27105  fsumdvdsmulOLD  27107  negsunif  27961  negsbdaylem  27962  onsiso  28169  bdayn0p1  28258  imaelshi  31987  vonf1owev  35095  cvmopnlem  35265  weiunfrlem  36452  ovoliunnfl  37656  voliunnfl  37658  volsupnfl  37659  gicabl  43088  permac8prim  45004