Theorem eqfnun 35160

Description: Two functions on 𝐴 ∪ 𝐵 are equal if and only if they have equal restrictions to both 𝐴 and 𝐵. (Contributed by Jeff Madsen, 19-Jun-2011.)

Assertion

Ref	Expression
eqfnun	⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))))

Proof of Theorem eqfnun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reseq1 5812	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
2		reseq1 5812	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))
3	1, 2	jca 515	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)))
4		elun 4076	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
5		fveq1 6644	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐺 ↾ 𝐴)‘𝑥))
6		fvres 6664	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
7	5, 6	sylan9req 2854	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
8		fvres 6664	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐺‘𝑥))
9	8	adantl 485	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐺‘𝑥))
10	7, 9	eqtr3d 2835	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
11	10	adantlr 714	. . . . . 6 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
12		fveq1 6644	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥))
13		fvres 6664	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
14	12, 13	sylan9req 2854	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
15		fvres 6664	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥))
16	15	adantl 485	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥))
17	14, 16	eqtr3d 2835	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
18	17	adantll 713	. . . . . 6 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
19	11, 18	jaodan 955	. . . . 5 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥))
20	4, 19	sylan2b 596	. . . 4 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥))
21	20	ralrimiva 3149	. . 3 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥))
22		eqfnfv 6779	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥)))
23	21, 22	syl5ibr 249	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → 𝐹 = 𝐺))
24	3, 23	impbid2 229	1 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∪ cun 3879   ↾ cres 5521   Fn wfn 6319  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332

This theorem is referenced by: (None)