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Mirrors > Home > MPE Home > Th. List > el2mpocl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
el2mpocl.o | ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) |
el2mpocl.e | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺)) |
Ref | Expression |
---|---|
el2mpocl | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el2mpocl.o | . . 3 ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) | |
2 | 1 | el2mpocsbcl 8109 | . 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 482 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴) | |
4 | simplr 769 | . . . . . . . 8 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵) | |
5 | el2mpocl.e | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺)) | |
6 | 5 | simpld 494 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹) |
7 | 6 | adantll 714 | . . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹) |
8 | 4, 7 | csbied 3946 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ⦋𝑌 / 𝑦⦌𝐶 = 𝐹) |
9 | 3, 8 | csbied 3946 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 = 𝐹) |
10 | 9 | eleq2d 2825 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ↔ 𝑆 ∈ 𝐹)) |
11 | 5 | simprd 495 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) |
12 | 11 | adantll 714 | . . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) |
13 | 4, 12 | csbied 3946 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ⦋𝑌 / 𝑦⦌𝐷 = 𝐺) |
14 | 3, 13 | csbied 3946 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 = 𝐺) |
15 | 14 | eleq2d 2825 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↔ 𝑇 ∈ 𝐺)) |
16 | 10, 15 | anbi12d 632 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷) ↔ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺))) |
17 | 16 | biimpd 229 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷) → (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺))) |
18 | 17 | imdistani 568 | . 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 35 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⦋csb 3908 (class class class)co 7431 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: wwlksonvtx 29885 wspthnonp 29889 |
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