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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 1121 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
3 | ovres 7524 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦)) | |
4 | 3 | adantl 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦)) |
5 | 2, 4 | eqtr4d 2776 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)) |
6 | 5 | ralrimivva 3194 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)) |
7 | eqid 2733 | . . 3 ⊢ (𝑌 × 𝑌) = (𝑌 × 𝑌) | |
8 | 6, 7 | jctil 521 | . 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 6673 | . . 3 ⊢ (𝜑 → 𝐹 Fn (𝑌 × 𝑌)) |
11 | oprres.g | . . . . 5 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆) | |
12 | 11 | ffnd 6673 | . . . 4 ⊢ (𝜑 → 𝐺 Fn (𝑋 × 𝑋)) |
13 | oprres.s | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
14 | xpss12 5652 | . . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
15 | 13, 13, 14 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
16 | fnssres 6628 | . . . 4 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) | |
17 | 12, 15, 16 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) |
18 | eqfnov 7489 | . . 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 585 | . 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ⊆ wss 3914 × cxp 5635 ↾ cres 5639 Fn wfn 6495 ⟶wf 6496 (class class class)co 7361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 |
This theorem is referenced by: subresre 40946 |
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