Theorem 2mpo0 7699

Description: If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.)

Hypotheses

Ref	Expression
2mpo0.o	⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)
2mpo0.u	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹))

Assertion

Ref	Expression
2mpo0	⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝐶,𝑠,𝑡   𝐷,𝑠,𝑡

Allowed substitution hints:   𝐴(𝑡,𝑠)   𝐵(𝑡,𝑠)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑡,𝑠)   𝑇(𝑥,𝑦,𝑡,𝑠)   𝐸(𝑥,𝑦,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑡,𝑠)   𝑋(𝑥,𝑦,𝑡,𝑠)   𝑌(𝑥,𝑦,𝑡,𝑠)

Proof of Theorem 2mpo0

Step	Hyp	Ref	Expression
1		ianor 982	. 2 ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) ↔ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∨ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)))
2		2mpo0.o	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)
3	2	mpondm0 7690	. . . . 5 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = ∅)
4	3	oveqd 7465	. . . 4 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆∅𝑇))
5		0ov 7485	. . . 4 ⊢ (𝑆∅𝑇) = ∅
6	4, 5	eqtrdi 2796	. . 3 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
7		notnotb 315	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ↔ ¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
8		2mpo0.u	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹))
9	8	adantr 480	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹))
10	9	oveqd 7465	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇))
11		eqid 2740	. . . . . . 7 ⊢ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)
12	11	mpondm0 7690	. . . . . 6 ⊢ (¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅)
13	12	adantl 481	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅)
14	10, 13	eqtrd 2780	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
15	7, 14	sylanbr 581	. . 3 ⊢ ((¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
16	6, 15	jaoi3 1061	. 2 ⊢ ((¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∨ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
17	1, 16	sylbi 217	1 ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∅c0 4352  (class class class)co 7448   ∈ cmpo 7450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-dm 5710  df-iota 6525  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453

This theorem is referenced by:  wwlksnon0  29887