Theorem iinfprg 49837

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 5893	. . . 4 ⊢ (𝑦 = 𝐴 → dom 𝑦 = dom 𝐴)
2		dmeq 5893	. . . 4 ⊢ (𝑦 = 𝐵 → dom 𝑦 = dom 𝐵)
3	1, 2	iinxprg 5055	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 = (dom 𝐴 ∩ dom 𝐵))
4		fveq1 6880	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦‘𝑥) = (𝐴‘𝑥))
5		fveq1 6880	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑥) = (𝐵‘𝑥))
6	4, 5	iinxprg 5055	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥) = ((𝐴‘𝑥) ∩ (𝐵‘𝑥)))
7	3, 6	mpteq12dv 5198	. 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 2769	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 400   = wceq 1570   ∈ wcel 2143   ∩ cin 3904  {cpr 4591  ∩ ciin 4957   ↦ cmpt 5192  dom cdm 5661  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-dm 5671  df-iota 6492  df-fv 6544

This theorem is referenced by:  infsubc  49838  infsubc2  49839