Theorem iinfprg 49721

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 5894	. . . 4 ⊢ (𝑦 = 𝐴 → dom 𝑦 = dom 𝐴)
2		dmeq 5894	. . . 4 ⊢ (𝑦 = 𝐵 → dom 𝑦 = dom 𝐵)
3	1, 2	iinxprg 5059	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 = (dom 𝐴 ∩ dom 𝐵))
4		fveq1 6881	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦‘𝑥) = (𝐴‘𝑥))
5		fveq1 6881	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑥) = (𝐵‘𝑥))
6	4, 5	iinxprg 5059	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥) = ((𝐴‘𝑥) ∩ (𝐵‘𝑥)))
7	3, 6	mpteq12dv 5202	. 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 2775	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 1567   ∈ wcel 2149   ∩ cin 3912  {cpr 4596  ∩ ciin 4961   ↦ cmpt 5196  dom cdm 5662  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-dm 5672  df-iota 6493  df-fv 6545

This theorem is referenced by:  infsubc  49722  infsubc2  49723