Theorem eqfnun 6978

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 5928	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
2		reseq1 5928	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))
3	1, 2	jca 511	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)))
4		elun 4102	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
5		fveq1 6829	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐺 ↾ 𝐴)‘𝑥))
6		fvres 6849	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
7	5, 6	sylan9req 2789	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
8		fvres 6849	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐺‘𝑥))
9	8	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐺‘𝑥))
10	7, 9	eqtr3d 2770	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
11	10	adantlr 715	. . . . . 6 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
12		fveq1 6829	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥))
13		fvres 6849	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
14	12, 13	sylan9req 2789	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
15		fvres 6849	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥))
16	15	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥))
17	14, 16	eqtr3d 2770	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
18	17	adantll 714	. . . . . 6 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
19	11, 18	jaodan 959	. . . . 5 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥))
20	4, 19	sylan2b 594	. . . 4 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥))
21	20	ralrimiva 3125	. . 3 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥))
22		eqfnfv 6972	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥)))
23	21, 22	imbitrrid 246	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → 𝐹 = 𝐺))
24	3, 23	impbid2 226	1 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ∪ cun 3896   ↾ cres 5623   Fn wfn 6483  ‘cfv 6488

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-fv 6496

This theorem is referenced by:  selvvvval  42706