Theorem 2mpt20 6780

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
2mpt20.o	⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)
2mpt20.u	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹))

Assertion

Ref	Expression
2mpt20	⊢ (¬ ((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 2mpt20

Step	Hyp	Ref	Expression
1		ianor 508	. 2 ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) ↔ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∨ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)))
2		2mpt20.o	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)
3	2	mpt2ndm0 6773	. . . . 5 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = ∅)
4	3	oveqd 6566	. . . 4 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆∅𝑇))
5		0ov 6580	. . . 4 ⊢ (𝑆∅𝑇) = ∅
6	4, 5	syl6eq 2660	. . 3 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
7		notnotb 303	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ↔ ¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
8		2mpt20.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 6566	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇))
11		eqid 2610	. . . . . . 7 ⊢ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)
12	11	mpt2ndm0 6773	. . . . . 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 2644	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
15	7, 14	sylanbr 489	. . 3 ⊢ ((¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
16	6, 15	jaoi3 1003	. 2 ⊢ ((¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∨ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
17	1, 16	sylbi 206	1 ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874  (class class class)co 6549   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  wwlksnon0  41123