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Theorem eqfnun 34989
Description: Two functions on 𝐴𝐵 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 (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵))))

Proof of Theorem eqfnun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 reseq1 5840 . . 3 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
2 reseq1 5840 . . 3 (𝐹 = 𝐺 → (𝐹𝐵) = (𝐺𝐵))
31, 2jca 514 . 2 (𝐹 = 𝐺 → ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)))
4 elun 4123 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 fveq1 6662 . . . . . . . . 9 ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐴)‘𝑥) = ((𝐺𝐴)‘𝑥))
6 fvres 6682 . . . . . . . . 9 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
75, 6sylan9req 2875 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐹𝑥))
8 fvres 6682 . . . . . . . . 9 (𝑥𝐴 → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
98adantl 484 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
107, 9eqtr3d 2856 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
1110adantlr 713 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
12 fveq1 6662 . . . . . . . . 9 ((𝐹𝐵) = (𝐺𝐵) → ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥))
13 fvres 6682 . . . . . . . . 9 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
1412, 13sylan9req 2875 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐹𝑥))
15 fvres 6682 . . . . . . . . 9 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1615adantl 484 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1714, 16eqtr3d 2856 . . . . . . 7 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1817adantll 712 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1911, 18jaodan 953 . . . . 5 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = (𝐺𝑥))
204, 19sylan2b 595 . . . 4 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝐹𝑥) = (𝐺𝑥))
2120ralrimiva 3180 . . 3 (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥))
22 eqfnfv 6795 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥)))
2321, 22syl5ibr 248 . 2 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → 𝐹 = 𝐺))
243, 23impbid2 228 1 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1530  wcel 2107  wral 3136  cun 3932  cres 5550   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by: (None)
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