Theorem eqfnun 7040

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 5973	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
2		reseq1 5973	. . 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 4144	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
5		fveq1 6890	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐺 ↾ 𝐴)‘𝑥))
6		fvres 6910	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
7	5, 6	sylan9req 2788	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
8		fvres 6910	. . . . . . . . 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 2769	. . . . . . 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 6890	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥))
13		fvres 6910	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
14	12, 13	sylan9req 2788	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
15		fvres 6910	. . . . . . . . 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 2769	. . . . . . 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 956	. . . . 5 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥))
20	4, 19	sylan2b 593	. . . 4 ⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥))
21	20	ralrimiva 3141	. . 3 ⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥))
22		eqfnfv 7034	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥)))
23	21, 22	imbitrrid 245	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → 𝐹 = 𝐺))
24	3, 23	impbid2 225	1 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099  ∀wral 3056   ∪ cun 3942   ↾ cres 5674   Fn wfn 6537  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550

This theorem is referenced by:  selvvvval  41740