Theorem fvreseq0 6357

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 6042	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
2		fnssres 6042	. . 3 ⊢ ((𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵) Fn 𝐵)
3		eqfnfv 6351	. . . 4 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐺 ↾ 𝐵) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥)))
4		fvres 6245	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
5		fvres 6245	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥))
6	4, 5	eqeq12d 2666	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
7	6	ralbiia 3008	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
8	3, 7	syl6bb 276	. . 3 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐺 ↾ 𝐵) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
9	1, 2, 8	syl2an 493	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
10	9	an4s 886	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607   ↾ cres 5145   Fn wfn 5921  ‘cfv 5926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934

This theorem is referenced by:  fvreseq1  6358  fvreseq  6359  limsupequzlem  40272