Theorem funimassov 5592

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 5167	. 2 ⊢ ((Fun 𝐹 ∧ (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶 ↔ ∀z ∈ (A × B)(𝐹‘z) ∈ 𝐶))
2		fveq2 5121	. . . . 5 ⊢ (z = ⟨x, y⟩ → (𝐹‘z) = (𝐹‘⟨x, y⟩))
3		df-ov 5458	. . . . 5 ⊢ (x𝐹y) = (𝐹‘⟨x, y⟩)
4	2, 3	syl6eqr 2087	. . . 4 ⊢ (z = ⟨x, y⟩ → (𝐹‘z) = (x𝐹y))
5	4	eleq1d 2103	. . 3 ⊢ (z = ⟨x, y⟩ → ((𝐹‘z) ∈ 𝐶 ↔ (x𝐹y) ∈ 𝐶))
6	5	ralxp 4422	. 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 1242   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911  ⟨cop 3370   × cxp 4286  dom cdm 4288   “ cima 4291  Fun wfun 4839  ‘cfv 4845  (class class class)co 5455

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-bnd 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-csb 2847  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-iun 3650  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  df-ov 5458

This theorem is referenced by: (None)