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Theorem oprres 7589
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 1118 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3 ovres 7587 . . . . . 6 ((𝑥𝑌𝑦𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
43adantl 481 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
52, 4eqtr4d 2771 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
65ralrimivva 3197 . . 3 (𝜑 → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
7 eqid 2728 . . 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 6723 . . 3 (𝜑𝐹 Fn (𝑌 × 𝑌))
11 oprres.g . . . . 5 (𝜑𝐺:(𝑋 × 𝑋)⟶𝑆)
1211ffnd 6723 . . . 4 (𝜑𝐺 Fn (𝑋 × 𝑋))
13 oprres.s . . . . 5 (𝜑𝑌𝑋)
14 xpss12 5693 . . . . 5 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
1513, 13, 14syl2anc 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))
1712, 15, 16syl2anc 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) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
1910, 17, 18syl2anc 583 . 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 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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