Theorem 2mpo0 7670

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 997	. 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 7661	. . . . 5 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = ∅)
4	3	oveqd 7437	. . . 4 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆∅𝑇))
5		0ov 7457	. . . 4 ⊢ (𝑆∅𝑇) = ∅
6	4, 5	eqtrdi 2812	. . 3 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
7		notnotb 318	. . . 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 486	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹))
10	9	oveqd 7437	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇))
11		eqid 2761	. . . . . . 7 ⊢ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)
12	11	mpondm0 7661	. . . . . 6 ⊢ (¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅)
13	12	adantl 487	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)𝑇) = ∅)
14	10, 13	eqtrd 2796	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
15	7, 14	sylanbr 594	. . 3 ⊢ ((¬ ¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
16	6, 15	jaoi3 1076	. 2 ⊢ ((¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∨ ¬ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
17	1, 16	sylbi 220	1 ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∅c0 4279  (class class class)co 7420   ∈ cmpo 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-dm 5661  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  wwlksnon0  30443