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Theorem resiun1OLD 5558
 Description: Obsolete proof of resiun1 5557 as of 25-Aug-2021. (Contributed by Mario Carneiro, 29-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
resiun1OLD ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem resiun1OLD
StepHypRef Expression
1 iunin2 4718 . 2 𝑥𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
2 df-res 5261 . . . . 5 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
3 incom 3956 . . . . 5 (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵)
42, 3eqtri 2793 . . . 4 (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵)
54a1i 11 . . 3 (𝑥𝐴 → (𝐵𝐶) = ((𝐶 × V) ∩ 𝐵))
65iuneq2i 4673 . 2 𝑥𝐴 (𝐵𝐶) = 𝑥𝐴 ((𝐶 × V) ∩ 𝐵)
7 df-res 5261 . . 3 ( 𝑥𝐴 𝐵𝐶) = ( 𝑥𝐴 𝐵 ∩ (𝐶 × V))
8 incom 3956 . . 3 ( 𝑥𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
97, 8eqtri 2793 . 2 ( 𝑥𝐴 𝐵𝐶) = ((𝐶 × V) ∩ 𝑥𝐴 𝐵)
101, 6, 93eqtr4ri 2804 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ∩ cin 3722  ∪ ciun 4654   × cxp 5247   ↾ cres 5251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-iun 4656  df-res 5261 This theorem is referenced by: (None)
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