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Mirrors > Home > MPE Home > Th. List > oprres | Structured version Visualization version GIF version |
Description: The restriction of an operation is an operation. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.) |
Ref | Expression |
---|---|
oprres.v | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
oprres.s | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
oprres.f | ⊢ (𝜑 → 𝐹:(𝑌 × 𝑌)⟶𝑅) |
oprres.g | ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆) |
Ref | Expression |
---|---|
oprres | ⊢ (𝜑 → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprres.v | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) | |
2 | 1 | 3expb 1118 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
3 | ovres 7587 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦)) | |
4 | 3 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦)) |
5 | 2, 4 | eqtr4d 2771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)) |
6 | 5 | ralrimivva 3197 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)) |
7 | eqid 2728 | . . 3 ⊢ (𝑌 × 𝑌) = (𝑌 × 𝑌) | |
8 | 6, 7 | jctil 519 | . 2 ⊢ (𝜑 → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))) |
9 | oprres.f | . . . 4 ⊢ (𝜑 → 𝐹:(𝑌 × 𝑌)⟶𝑅) | |
10 | 9 | ffnd 6723 | . . 3 ⊢ (𝜑 → 𝐹 Fn (𝑌 × 𝑌)) |
11 | oprres.g | . . . . 5 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆) | |
12 | 11 | ffnd 6723 | . . . 4 ⊢ (𝜑 → 𝐺 Fn (𝑋 × 𝑋)) |
13 | oprres.s | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
14 | xpss12 5693 | . . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
15 | 13, 13, 14 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
16 | fnssres 6678 | . . . 4 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) | |
17 | 12, 15, 16 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) |
18 | eqfnov 7550 | . . 3 ⊢ ((𝐹 Fn (𝑌 × 𝑌) ∧ (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))) | |
19 | 10, 17, 18 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))) |
20 | 8, 19 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ⊆ wss 3947 × cxp 5676 ↾ cres 5680 Fn wfn 6543 ⟶wf 6544 (class class class)co 7420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 |
This theorem is referenced by: subresre 41985 |
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