Theorem bnj1536 34331

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 7041	. . 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 257	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540  ∀wral 3060   ⊆ wss 3948   ↾ cres 5678   Fn wfn 6538  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  bnj1523  34548