Theorem iinfprg 49091

Description: Indexed intersection of functions with an unordered pair index. (Contributed by Zhi Wang, 31-Oct-2025.)

Assertion

Ref	Expression
iinfprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) = (𝑥 ∈ ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 ↦ ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉   𝑥,𝑊

Allowed substitution hints:   𝑉(𝑦)   𝑊(𝑦)

Proof of Theorem iinfprg

Step	Hyp	Ref	Expression
1		dmeq 5838	. . . 4 ⊢ (𝑦 = 𝐴 → dom 𝑦 = dom 𝐴)
2		dmeq 5838	. . . 4 ⊢ (𝑦 = 𝐵 → dom 𝑦 = dom 𝐵)
3	1, 2	iinxprg 5032	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 = (dom 𝐴 ∩ dom 𝐵))
4		fveq1 6816	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦‘𝑥) = (𝐴‘𝑥))
5		fveq1 6816	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑥) = (𝐵‘𝑥))
6	4, 5	iinxprg 5032	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥) = ((𝐴‘𝑥) ∩ (𝐵‘𝑥)))
7	3, 6	mpteq12dv 5173	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 ↦ ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥)) = (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))))
8	7	eqcomd 2737	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) = (𝑥 ∈ ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 ↦ ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∩ cin 3896  {cpr 4573  ∩ ciin 4937   ↦ cmpt 5167  dom cdm 5611  ‘cfv 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-dm 5621  df-iota 6432  df-fv 6484

This theorem is referenced by:  infsubc  49092  infsubc2  49093