Theorem oprres 7514

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	⊢ ((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)

Assertion

Ref	Expression
oprres	⊢ (𝜑 → 𝐹 = (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

Step	Hyp	Ref	Expression
1		oprres.v	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
2	1	3expb 1120	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3		ovres 7512	. . . . . 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 2769	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
6	5	ralrimivva 3175	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
7		eqid 2731	. . 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 6652	. . 3 ⊢ (𝜑 → 𝐹 Fn (𝑌 × 𝑌))
11		oprres.g	. . . . 5 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆)
12	11	ffnd 6652	. . . 4 ⊢ (𝜑 → 𝐺 Fn (𝑋 × 𝑋))
13		oprres.s	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
14		xpss12 5629	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
15	13, 13, 14	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
16		fnssres 6604	. . . 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 7475	. . 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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897   × cxp 5612   ↾ cres 5616   Fn wfn 6476  ⟶wf 6477  (class class class)co 7346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349

This theorem is referenced by:  subresre  42534