Theorem ralima 7214

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 6621	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	funfnd 6550	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹)
3		fndm 6624	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	sseq2d 3982	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
5	4	biimpar 477	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
6		fvexd 6876	. . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V)
7		fvelimab 6936	. . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
8		eqcom 2737	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
9	8	rexbii 3077	. . . 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 5368	. 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 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  dom cdm 5641   “ cima 5644   Fn wfn 6509  ‘cfv 6514

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  rexima  7215  supisolem  9432  ordtypelem6  9483  ordtypelem7  9484  limsupgle  15450  mrcuni  17589  ipodrsima  18507  mgmhmima  18649  mhmimalem  18758  ghmnsgima  19179  cntzmhm  19280  rhmimasubrnglem  20481  qtopeu  23610  kqdisj  23626  ghmcnp  24009  qustgplem  24015  qtopbaslem  24653  bndth  24864  fmcfil  25179  ovoliunlem1  25410  volsup2  25513  mbflimsup  25574  itg2gt0  25668  mdegleb  25976  efopn  26574  fsumdvdsmul  27112  fsumdvdsmulOLD  27114  negsunif  27968  negsbdaylem  27969  onsiso  28176  bdayn0p1  28265  imaelshi  31994  vonf1owev  35102  cvmopnlem  35272  weiunfrlem  36459  ovoliunnfl  37663  voliunnfl  37665  volsupnfl  37666  gicabl  43095  permac8prim  45011