Theorem iinfprg 50281

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 5886	. . . 4 ⊢ (𝑦 = 𝐴 → dom 𝑦 = dom 𝐴)
2		dmeq 5886	. . . 4 ⊢ (𝑦 = 𝐵 → dom 𝑦 = dom 𝐵)
3	1, 2	iinxprg 5048	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 = (dom 𝐴 ∩ dom 𝐵))
4		fveq1 6889	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦‘𝑥) = (𝐴‘𝑥))
5		fveq1 6889	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑥) = (𝐵‘𝑥))
6	4, 5	iinxprg 5048	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥) = ((𝐴‘𝑥) ∩ (𝐵‘𝑥)))
7	3, 6	mpteq12dv 5191	. 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 2766	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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∩ cin 4088  {cpr 4586  ∩ ciin 4951   ↦ cmpt 5185  dom cdm 5651  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-dm 5661  df-iota 6497  df-fv 6551

This theorem is used by:  infsubc  50282  infsubc2  50283