Theorem bnj1536 32834

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1536.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
bnj1536.2	⊢ (𝜑 → 𝐺 Fn 𝐴)
bnj1536.3	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
bnj1536.4	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))

Assertion

Ref	Expression
bnj1536	⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))

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

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

Proof of Theorem bnj1536

Step	Hyp	Ref	Expression
1		bnj1536.4	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
2		bnj1536.1	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		bnj1536.2	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
4		bnj1536.3	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
5		fvreseq 6917	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
6	2, 3, 4, 5	syl21anc 835	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
7	1, 6	mpbird 256	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ∀wral 3064   ⊆ wss 3887   ↾ cres 5591   Fn wfn 6428  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  bnj1523  33051