Theorem oprres 7578

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 1138	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3		ovres 7576	. . . . . 6 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
4	3	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
5	2, 4	eqtr4d 2801	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
6	5	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
7		eqid 2763	. . 3 ⊢ (𝑌 × 𝑌) = (𝑌 × 𝑌)
8	6, 7	jctil 528	. 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 6706	. . 3 ⊢ (𝜑 → 𝐹 Fn (𝑌 × 𝑌))
11		oprres.g	. . . . 5 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆)
12	11	ffnd 6706	. . . 4 ⊢ (𝜑 → 𝐺 Fn (𝑋 × 𝑋))
13		oprres.s	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
14		xpss12 5676	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
15	13, 13, 14	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
16		fnssres 6658	. . . 4 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
17	12, 15, 16	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
18		eqfnov 7539	. . 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 595	. 2 ⊢ (𝜑 → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
20	8, 19	mpbird 260	1 ⊢ (𝜑 → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905   × cxp 5659   ↾ cres 5663   Fn wfn 6531  ⟶wf 6532  (class class class)co 7410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413

This theorem is used by:  subresre  43220