Theorem fvreseq0 7035

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 6660	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
2		fnssres 6660	. . 3 ⊢ ((𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵) Fn 𝐵)
3		eqfnfv 7027	. . . 4 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐺 ↾ 𝐵) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥)))
4		fvres 6902	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
5		fvres 6902	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥))
6	4, 5	eqeq12d 2779	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
7	6	ralbiia 3109	. . . 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 607	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
10	9	an4s 672	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 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3906   ↾ cres 5665   Fn wfn 6533  ‘cfv 6538

This theorem was proved from 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 5258  ax-nul 5270  ax-pr 5406

This theorem 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 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is referenced by:  fvreseq1  7036  fvreseq  7037  ply1degltdimlem  33993  limsupequzlem  46419