Theorem fvreseq0 7034

Description: Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.)

Assertion

Ref	Expression
fvreseq0	⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem fvreseq0

Step	Hyp	Ref	Expression
1		fnssres 6659	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
2		fnssres 6659	. . 3 ⊢ ((𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵) Fn 𝐵)
3		eqfnfv 7026	. . . 4 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐺 ↾ 𝐵) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥)))
4		fvres 6901	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
5		fvres 6901	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥))
6	4, 5	eqeq12d 2778	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
7	6	ralbiia 3108	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
8	3, 7	bitrdi 290	. . 3 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐺 ↾ 𝐵) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
9	1, 2, 8	syl2an 608	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
10	9	an4s 673	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902   ↾ cres 5661   Fn wfn 6532  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is used by:  fvreseq1  7035  fvreseq  7036  ply1degltdimlem  34140  limsupequzlem  46558