MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprres Structured version   Visualization version   GIF version

Theorem oprres 7526
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 7524 . . . . . 6 ((𝑥𝑌𝑦𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
43adantl 483 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
52, 4eqtr4d 2776 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
65ralrimivva 3194 . . 3 (𝜑 → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
7 eqid 2733 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
86, 7jctil 521 . 2 (𝜑 → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
9 oprres.f . . . 4 (𝜑𝐹:(𝑌 × 𝑌)⟶𝑅)
109ffnd 6673 . . 3 (𝜑𝐹 Fn (𝑌 × 𝑌))
11 oprres.g . . . . 5 (𝜑𝐺:(𝑋 × 𝑋)⟶𝑆)
1211ffnd 6673 . . . 4 (𝜑𝐺 Fn (𝑋 × 𝑋))
13 oprres.s . . . . 5 (𝜑𝑌𝑋)
14 xpss12 5652 . . . . 5 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
1513, 13, 14syl2anc 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))
1712, 15, 16syl2anc 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) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
1910, 17, 18syl2anc 585 . 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 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
  Copyright terms: Public domain W3C validator