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Theorem iinfprg 49628
Description: Indexed intersection of functions with an unordered pair index. (Contributed by Zhi Wang, 31-Oct-2025.)
Assertion
Ref Expression
iinfprg ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴𝑥) ∩ (𝐵𝑥))) = (𝑥 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 𝑦 ∈ {𝐴, 𝐵} (𝑦𝑥)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉   𝑥,𝑊
Allowed substitution hints:   𝑉(𝑦)   𝑊(𝑦)

Proof of Theorem iinfprg
StepHypRef Expression
1 dmeq 5872 . . . 4 (𝑦 = 𝐴 → dom 𝑦 = dom 𝐴)
2 dmeq 5872 . . . 4 (𝑦 = 𝐵 → dom 𝑦 = dom 𝐵)
31, 2iinxprg 5040 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 = (dom 𝐴 ∩ dom 𝐵))
4 fveq1 6855 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥) = (𝐴𝑥))
5 fveq1 6855 . . . 4 (𝑦 = 𝐵 → (𝑦𝑥) = (𝐵𝑥))
64, 5iinxprg 5040 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑦 ∈ {𝐴, 𝐵} (𝑦𝑥) = ((𝐴𝑥) ∩ (𝐵𝑥)))
73, 6mpteq12dv 5181 . 2 ((𝐴𝑉𝐵𝑊) → (𝑥 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 𝑦 ∈ {𝐴, 𝐵} (𝑦𝑥)) = (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴𝑥) ∩ (𝐵𝑥))))
87eqcomd 2762 1 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴𝑥) ∩ (𝐵𝑥))) = (𝑥 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 𝑦 ∈ {𝐴, 𝐵} (𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1554  wcel 2136  cin 3898  {cpr 4578   ciin 4944  cmpt 5175  dom cdm 5640  cfv 6510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iin 4946  df-br 5095  df-opab 5157  df-mpt 5176  df-dm 5650  df-iota 6466  df-fv 6518
This theorem is referenced by:  infsubc  49629  infsubc2  49630
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