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Mirrors > Home > MPE Home > Th. List > el2mpt2cl | 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 |
---|---|
el2mpt2cl.o | ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) |
el2mpt2cl.e | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺)) |
Ref | Expression |
---|---|
el2mpt2cl | ⊢ (∀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 | el2mpt2cl.o | . . 3 ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) | |
2 | 1 | el2mpt2csbcl 7250 | . 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 473 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴) | |
4 | simplr 792 | . . . . . . . 8 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵) | |
5 | el2mpt2cl.e | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐶 = 𝐹 ∧ 𝐷 = 𝐺)) | |
6 | 5 | simpld 475 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹) |
7 | 6 | adantll 750 | . . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐶 = 𝐹) |
8 | 4, 7 | csbied 3560 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ⦋𝑌 / 𝑦⦌𝐶 = 𝐹) |
9 | 3, 8 | csbied 3560 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 = 𝐹) |
10 | 9 | eleq2d 2687 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ↔ 𝑆 ∈ 𝐹)) |
11 | 5 | simprd 479 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) |
12 | 11 | adantll 750 | . . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) |
13 | 4, 12 | csbied 3560 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ⦋𝑌 / 𝑦⦌𝐷 = 𝐺) |
14 | 3, 13 | csbied 3560 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 = 𝐺) |
15 | 14 | eleq2d 2687 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↔ 𝑇 ∈ 𝐺)) |
16 | 10, 15 | anbi12d 747 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷) ↔ (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺))) |
17 | 16 | biimpd 219 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷) → (𝑆 ∈ 𝐹 ∧ 𝑇 ∈ 𝐺))) |
18 | 17 | imdistani 726 | . 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 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⦋csb 3533 (class class class)co 6650 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: wspthnonp 26744 |
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