Theorem rexima 5337

Description: Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypothesis

Ref	Expression
rexima.x	⊢ (x = (𝐹‘y) → (φ ↔ ψ))

Assertion

Ref	Expression
rexima	⊢ ((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   x,A,y

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem rexima

Step	Hyp	Ref	Expression
1		ssel2 2934	. . . 4 ⊢ ((B ⊆ A ∧ y ∈ B) → y ∈ A)
2		funfvex 5135	. . . . 5 ⊢ ((Fun 𝐹 ∧ y ∈ dom 𝐹) → (𝐹‘y) ∈ V)
3	2	funfni 4942	. . . 4 ⊢ ((𝐹 Fn A ∧ y ∈ A) → (𝐹‘y) ∈ V)
4	1, 3	sylan2 270	. . 3 ⊢ ((𝐹 Fn A ∧ (B ⊆ A ∧ y ∈ B)) → (𝐹‘y) ∈ V)
5	4	anassrs 380	. 2 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ y ∈ B) → (𝐹‘y) ∈ V)
6		fvelimab 5172	. . 3 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (x ∈ (𝐹 “ B) ↔ ∃y ∈ B (𝐹‘y) = x))
7		eqcom 2039	. . . 4 ⊢ ((𝐹‘y) = x ↔ x = (𝐹‘y))
8	7	rexbii 2325	. . 3 ⊢ (∃y ∈ B (𝐹‘y) = x ↔ ∃y ∈ B x = (𝐹‘y))
9	6, 8	syl6bb 185	. 2 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (x ∈ (𝐹 “ B) ↔ ∃y ∈ B x = (𝐹‘y)))
10		rexima.x	. . 3 ⊢ (x = (𝐹‘y) → (φ ↔ ψ))
11	10	adantl 262	. 2 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ x = (𝐹‘y)) → (φ ↔ ψ))
12	5, 9, 11	rexxfr2d 4163	1 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃x ∈ (𝐹 “ B)φ ↔ ∃y ∈ B ψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  Vcvv 2551   ⊆ wss 2911   “ cima 4291   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by: (None)