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Theorem oprres 7601
Description: The restriction of an operation is an operation. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.)
Hypotheses
Ref Expression
oprres.v ((𝜑𝑥𝑌𝑦𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
oprres.s (𝜑𝑌𝑋)
oprres.f (𝜑𝐹:(𝑌 × 𝑌)⟶𝑅)
oprres.g (𝜑𝐺:(𝑋 × 𝑋)⟶𝑆)
Assertion
Ref Expression
oprres (𝜑𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem oprres
StepHypRef Expression
1 oprres.v . . . . . 6 ((𝜑𝑥𝑌𝑦𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
213expb 1121 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3 ovres 7599 . . . . . 6 ((𝑥𝑌𝑦𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
43adantl 481 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
52, 4eqtr4d 2780 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
65ralrimivva 3202 . . 3 (𝜑 → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
7 eqid 2737 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
86, 7jctil 519 . 2 (𝜑 → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
9 oprres.f . . . 4 (𝜑𝐹:(𝑌 × 𝑌)⟶𝑅)
109ffnd 6737 . . 3 (𝜑𝐹 Fn (𝑌 × 𝑌))
11 oprres.g . . . . 5 (𝜑𝐺:(𝑋 × 𝑋)⟶𝑆)
1211ffnd 6737 . . . 4 (𝜑𝐺 Fn (𝑋 × 𝑋))
13 oprres.s . . . . 5 (𝜑𝑌𝑋)
14 xpss12 5700 . . . . 5 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
1513, 13, 14syl2anc 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))
1712, 15, 16syl2anc 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) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
1910, 17, 18syl2anc 584 . 2 (𝜑 → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
208, 19mpbird 257 1 (𝜑𝐹 = (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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