Theorem bnj1536 35151

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 7023	. . 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 848	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
7	1, 6	mpbird 259	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562  ∀wral 3078   ⊆ wss 3906   ↾ cres 5651   Fn wfn 6518  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531

This theorem is referenced by:  bnj1523  35368