Theorem ralima 7192

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 6598	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	funfnd 6529	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹)
3		fndm 6601	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	sseq2d 3954	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
5	4	biimpar 477	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
6		fvexd 6855	. . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V)
7		fvelimab 6912	. . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
8		eqcom 2743	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
9	8	rexbii 3084	. . . 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 5352	. 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
14	2, 5, 13	syl2an2r 686	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  dom cdm 5631   “ cima 5634   Fn wfn 6493  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506

This theorem is referenced by:  rexima  7193  supisolem  9387  ordtypelem6  9438  ordtypelem7  9439  limsupgle  15439  mrcuni  17587  ipodrsima  18507  mgmhmima  18683  mhmimalem  18792  ghmnsgima  19215  cntzmhm  19316  rhmimasubrnglem  20542  qtopeu  23681  kqdisj  23697  ghmcnp  24080  qustgplem  24086  qtopbaslem  24723  bndth  24925  fmcfil  25239  ovoliunlem1  25469  volsup2  25572  mbflimsup  25633  itg2gt0  25727  mdegleb  26029  efopn  26622  fsumdvdsmul  27158  negsunif  28047  negbdaylem  28048  oniso  28263  bdayn0p1  28361  imaelshi  32129  vonf1owev  35290  cvmopnlem  35460  weiunfrlem  36646  ovoliunnfl  37983  voliunnfl  37985  volsupnfl  37986  gicabl  43527  permac8prim  45441