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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 7599 | . . . . . 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 2780 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)) | 
| 6 | 5 | ralrimivva 3202 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)) | 
| 7 | eqid 2737 | . . 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 6737 | . . 3 ⊢ (𝜑 → 𝐹 Fn (𝑌 × 𝑌)) | 
| 11 | oprres.g | . . . . 5 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆) | |
| 12 | 11 | ffnd 6737 | . . . 4 ⊢ (𝜑 → 𝐺 Fn (𝑋 × 𝑋)) | 
| 13 | oprres.s | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
| 14 | xpss12 5700 | . . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
| 15 | 13, 13, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | 
| 16 | fnssres 6691 | . . . 4 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) | |
| 17 | 12, 15, 16 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) | 
| 18 | eqfnov 7562 | . . 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 584 | . 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 × cxp 5683 ↾ cres 5687 Fn wfn 6556 ⟶wf 6557 (class class class)co 7431 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: subresre 42460 | 
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