Theorem rexima 5319

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 2917	. . . 4 ⊢ ((B ⊆ A ∧ y ∈ B) → y ∈ A)
2		funfvex 5117	. . . . 5 ⊢ ((Fun 𝐹 ∧ y ∈ dom 𝐹) → (𝐹‘y) ∈ V)
3	2	funfni 4925	. . . 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 382	. 2 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ y ∈ B) → (𝐹‘y) ∈ V)
6		fvelimab 5154	. . 3 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (x ∈ (𝐹 “ B) ↔ ∃y ∈ B (𝐹‘y) = x))
7		eqcom 2024	. . . 4 ⊢ ((𝐹‘y) = x ↔ x = (𝐹‘y))
8	7	rexbii 2309	. . 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 4147	1 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃x ∈ (𝐹 “ B)φ ↔ ∃y ∈ B ψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  Vcvv 2535   ⊆ wss 2894   “ cima 4275   Fn wfn 4824  ‘cfv 4829

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837

This theorem is referenced by: (None)