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Theorem inimasn 4668
Description: The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
inimasn (𝐶 𝑉 → ((AB) “ {𝐶}) = ((A “ {𝐶}) ∩ (B “ {𝐶})))

Proof of Theorem inimasn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3103 . . 3 (x ((A “ {𝐶}) ∩ (B “ {𝐶})) ↔ (x (A “ {𝐶}) x (B “ {𝐶})))
2 elin 3103 . . . . 5 (⟨𝐶, x (AB) ↔ (⟨𝐶, x A 𝐶, x B))
32a1i 9 . . . 4 (𝐶 𝑉 → (⟨𝐶, x (AB) ↔ (⟨𝐶, x A 𝐶, x B)))
4 vex 2538 . . . . 5 x V
5 elimasng 4620 . . . . 5 ((𝐶 𝑉 x V) → (x ((AB) “ {𝐶}) ↔ ⟨𝐶, x (AB)))
64, 5mpan2 403 . . . 4 (𝐶 𝑉 → (x ((AB) “ {𝐶}) ↔ ⟨𝐶, x (AB)))
7 elimasng 4620 . . . . . 6 ((𝐶 𝑉 x V) → (x (A “ {𝐶}) ↔ ⟨𝐶, x A))
84, 7mpan2 403 . . . . 5 (𝐶 𝑉 → (x (A “ {𝐶}) ↔ ⟨𝐶, x A))
9 elimasng 4620 . . . . . 6 ((𝐶 𝑉 x V) → (x (B “ {𝐶}) ↔ ⟨𝐶, x B))
104, 9mpan2 403 . . . . 5 (𝐶 𝑉 → (x (B “ {𝐶}) ↔ ⟨𝐶, x B))
118, 10anbi12d 445 . . . 4 (𝐶 𝑉 → ((x (A “ {𝐶}) x (B “ {𝐶})) ↔ (⟨𝐶, x A 𝐶, x B)))
123, 6, 113bitr4rd 210 . . 3 (𝐶 𝑉 → ((x (A “ {𝐶}) x (B “ {𝐶})) ↔ x ((AB) “ {𝐶})))
131, 12syl5rbb 182 . 2 (𝐶 𝑉 → (x ((AB) “ {𝐶}) ↔ x ((A “ {𝐶}) ∩ (B “ {𝐶}))))
1413eqrdv 2020 1 (𝐶 𝑉 → ((AB) “ {𝐶}) = ((A “ {𝐶}) ∩ (B “ {𝐶})))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  cin 2893  {csn 3350  cop 3353  cima 4275
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-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285
This theorem is referenced by: (None)
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