Theorem el2mpocl 8077

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. Using implicit substitution. (Contributed by AV, 21-May-2021.)

Hypotheses

Ref	Expression
el2mpocl.o	⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸))
el2mpocl.e	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺))

Assertion

Ref	Expression
el2mpocl	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺))))

Distinct variable groups:   𝐴,𝑠,𝑡,𝑥,𝑦   𝐵,𝑠,𝑡,𝑥,𝑦   𝐶,𝑠,𝑡   𝐷,𝑠,𝑡   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑋,𝑠,𝑡,𝑥,𝑦   𝑌,𝑠,𝑡,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑡,𝑠)   𝑇(𝑥,𝑦,𝑡,𝑠)   𝑈(𝑡,𝑠)   𝐸(𝑥,𝑦,𝑡,𝑠)   𝐹(𝑡,𝑠)   𝐺(𝑡,𝑠)   𝑂(𝑥,𝑦,𝑡,𝑠)   𝑉(𝑡,𝑠)   𝑊(𝑥,𝑦,𝑡,𝑠)

Proof of Theorem el2mpocl

Step	Hyp	Ref	Expression
1		el2mpocl.o	. . 3 ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸))
2	1	el2mpocsbcl 8076	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))))
3		simpl 487	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
4		simplr 780	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵)
5		el2mpocl.e	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺))
6	5	simpld 499	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹)
7	6	adantll 726	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹)
8	4, 7	csbied 3889	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ⦋𝑌 / 𝑦⦌𝐶 = 𝐹)
9	3, 8	csbied 3889	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 = 𝐹)
10	9	eleq2d 2849	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ↔ 𝑆 ∈ 𝐹))
11	5	simprd 500	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
12	11	adantll 726	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
13	4, 12	csbied 3889	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ⦋𝑌 / 𝑦⦌𝐷 = 𝐺)
14	3, 13	csbied 3889	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 = 𝐺)
15	14	eleq2d 2849	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↔ 𝑇 ∈ 𝐺))
16	10, 15	anbi12d 643	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷) ↔ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺)))
17	16	biimpd 232	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷) → (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺)))
18	17	imdistani 578	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺)))
19	2, 18	syl6 36	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⦋csb 3853  (class class class)co 7410   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983

This theorem is referenced by:  wwlksonvtx  30204  wspthnonp  30208