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Theorem oprres 7335
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 7333 . . . . . 6 ((𝑥𝑌𝑦𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
43adantl 485 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
52, 4eqtr4d 2777 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
65ralrimivva 3104 . . 3 (𝜑 → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
7 eqid 2739 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
86, 7jctil 523 . 2 (𝜑 → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
9 oprres.f . . . 4 (𝜑𝐹:(𝑌 × 𝑌)⟶𝑅)
109ffnd 6506 . . 3 (𝜑𝐹 Fn (𝑌 × 𝑌))
11 oprres.g . . . . 5 (𝜑𝐺:(𝑋 × 𝑋)⟶𝑆)
1211ffnd 6506 . . . 4 (𝜑𝐺 Fn (𝑋 × 𝑋))
13 oprres.s . . . . 5 (𝜑𝑌𝑋)
14 xpss12 5541 . . . . 5 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
1513, 13, 14syl2anc 587 . . . 4 (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
16 fnssres 6460 . . . 4 ((𝐺 Fn (𝑋 × 𝑋) ∧ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
1712, 15, 16syl2anc 587 . . 3 (𝜑 → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
18 eqfnov 7298 . . 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 587 . 2 (𝜑 → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
208, 19mpbird 260 1 (𝜑𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wss 3844   × cxp 5524  cres 5528   Fn wfn 6335  wf 6336  (class class class)co 7173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7176
This theorem is referenced by:  subresre  40012
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