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Theorem inimasn 4684
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 3120 . . 3 (x ((A “ {𝐶}) ∩ (B “ {𝐶})) ↔ (x (A “ {𝐶}) x (B “ {𝐶})))
2 elin 3120 . . . . 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 2554 . . . . 5 x V
5 elimasng 4636 . . . . 5 ((𝐶 𝑉 x V) → (x ((AB) “ {𝐶}) ↔ ⟨𝐶, x (AB)))
64, 5mpan2 401 . . . 4 (𝐶 𝑉 → (x ((AB) “ {𝐶}) ↔ ⟨𝐶, x (AB)))
7 elimasng 4636 . . . . . 6 ((𝐶 𝑉 x V) → (x (A “ {𝐶}) ↔ ⟨𝐶, x A))
84, 7mpan2 401 . . . . 5 (𝐶 𝑉 → (x (A “ {𝐶}) ↔ ⟨𝐶, x A))
9 elimasng 4636 . . . . . 6 ((𝐶 𝑉 x V) → (x (B “ {𝐶}) ↔ ⟨𝐶, x B))
104, 9mpan2 401 . . . . 5 (𝐶 𝑉 → (x (B “ {𝐶}) ↔ ⟨𝐶, x B))
118, 10anbi12d 442 . . . 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 2035 1 (𝐶 𝑉 → ((AB) “ {𝐶}) = ((A “ {𝐶}) ∩ (B “ {𝐶})))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  {csn 3367  cop 3370  cima 4291
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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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