Theorem funimassov 5573

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)

Assertion

Ref	Expression
funimassov	⊢ ((Fun 𝐹 ∧ (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶 ↔ ∀x ∈ A ∀y ∈ B (x𝐹y) ∈ 𝐶))

Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y   x,𝐹,y

Proof of Theorem funimassov

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimass4 5149	. 2 ⊢ ((Fun 𝐹 ∧ (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶 ↔ ∀z ∈ (A × B)(𝐹‘z) ∈ 𝐶))
2		fveq2 5103	. . . . 5 ⊢ (z = ⟨x, y⟩ → (𝐹‘z) = (𝐹‘⟨x, y⟩))
3		df-ov 5439	. . . . 5 ⊢ (x𝐹y) = (𝐹‘⟨x, y⟩)
4	2, 3	syl6eqr 2072	. . . 4 ⊢ (z = ⟨x, y⟩ → (𝐹‘z) = (x𝐹y))
5	4	eleq1d 2088	. . 3 ⊢ (z = ⟨x, y⟩ → ((𝐹‘z) ∈ 𝐶 ↔ (x𝐹y) ∈ 𝐶))
6	5	ralxp 4406	. 2 ⊢ (∀z ∈ (A × B)(𝐹‘z) ∈ 𝐶 ↔ ∀x ∈ A ∀y ∈ B (x𝐹y) ∈ 𝐶)
7	1, 6	syl6bb 185	1 ⊢ ((Fun 𝐹 ∧ (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶 ↔ ∀x ∈ A ∀y ∈ B (x𝐹y) ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∀wral 2284   ⊆ wss 2894  ⟨cop 3353   × cxp 4270  dom cdm 4272   “ cima 4275  Fun wfun 4823  ‘cfv 4829  (class class class)co 5436

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-csb 2830  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-iun 3633  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  df-ov 5439

This theorem is referenced by: (None)