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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)) ⊆ 𝐶x 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.
StepHypRef Expression
1 funimass4 5149 . 2 ((Fun 𝐹 (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶z (A × B)(𝐹z) 𝐶))
2 fveq2 5103 . . . . 5 (z = ⟨x, y⟩ → (𝐹z) = (𝐹‘⟨x, y⟩))
3 df-ov 5439 . . . . 5 (x𝐹y) = (𝐹‘⟨x, y⟩)
42, 3syl6eqr 2072 . . . 4 (z = ⟨x, y⟩ → (𝐹z) = (x𝐹y))
54eleq1d 2088 . . 3 (z = ⟨x, y⟩ → ((𝐹z) 𝐶 ↔ (x𝐹y) 𝐶))
65ralxp 4406 . 2 (z (A × B)(𝐹z) 𝐶x A y B (x𝐹y) 𝐶)
71, 6syl6bb 185 1 ((Fun 𝐹 (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶x A y B (x𝐹y) 𝐶))
Colors of variables: wff set class
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)
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