Theorem bnj1536 30012

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 6212	. . 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 1316	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
7	1, 6	mpbird 245	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))

Syntax hints:   → wi 4   ↔ wb 194   = wceq 1474  ∀wral 2895   ⊆ wss 3539   ↾ cres 5030   Fn wfn 5785  ‘cfv 5790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798

This theorem is referenced by:  bnj1523  30227