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Theorem eqfnun 35888
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 5878 . . 3 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
2 reseq1 5878 . . 3 (𝐹 = 𝐺 → (𝐹𝐵) = (𝐺𝐵))
31, 2jca 512 . 2 (𝐹 = 𝐺 → ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)))
4 elun 4082 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 fveq1 6765 . . . . . . . . 9 ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐴)‘𝑥) = ((𝐺𝐴)‘𝑥))
6 fvres 6785 . . . . . . . . 9 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
75, 6sylan9req 2799 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐹𝑥))
8 fvres 6785 . . . . . . . . 9 (𝑥𝐴 → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
98adantl 482 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
107, 9eqtr3d 2780 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
1110adantlr 712 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
12 fveq1 6765 . . . . . . . . 9 ((𝐹𝐵) = (𝐺𝐵) → ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥))
13 fvres 6785 . . . . . . . . 9 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
1412, 13sylan9req 2799 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐹𝑥))
15 fvres 6785 . . . . . . . . 9 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1615adantl 482 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1714, 16eqtr3d 2780 . . . . . . 7 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1817adantll 711 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1911, 18jaodan 955 . . . . 5 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = (𝐺𝑥))
204, 19sylan2b 594 . . . 4 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝐹𝑥) = (𝐺𝑥))
2120ralrimiva 3108 . . 3 (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥))
22 eqfnfv 6901 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥)))
2321, 22syl5ibr 245 . 2 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → 𝐹 = 𝐺))
243, 23impbid2 225 1 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  wral 3064  cun 3884  cres 5586   Fn wfn 6421  cfv 6426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-fv 6434
This theorem is referenced by: (None)
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