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Theorem eqfnun 7057
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 5991 . . 3 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
2 reseq1 5991 . . 3 (𝐹 = 𝐺 → (𝐹𝐵) = (𝐺𝐵))
31, 2jca 511 . 2 (𝐹 = 𝐺 → ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)))
4 elun 4153 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 fveq1 6905 . . . . . . . . 9 ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐴)‘𝑥) = ((𝐺𝐴)‘𝑥))
6 fvres 6925 . . . . . . . . 9 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
75, 6sylan9req 2798 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐹𝑥))
8 fvres 6925 . . . . . . . . 9 (𝑥𝐴 → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
98adantl 481 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
107, 9eqtr3d 2779 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
1110adantlr 715 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
12 fveq1 6905 . . . . . . . . 9 ((𝐹𝐵) = (𝐺𝐵) → ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥))
13 fvres 6925 . . . . . . . . 9 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
1412, 13sylan9req 2798 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐹𝑥))
15 fvres 6925 . . . . . . . . 9 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1615adantl 481 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1714, 16eqtr3d 2779 . . . . . . 7 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1817adantll 714 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1911, 18jaodan 960 . . . . 5 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = (𝐺𝑥))
204, 19sylan2b 594 . . . 4 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝐹𝑥) = (𝐺𝑥))
2120ralrimiva 3146 . . 3 (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥))
22 eqfnfv 7051 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥)))
2321, 22imbitrrid 246 . 2 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → 𝐹 = 𝐺))
243, 23impbid2 226 1 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  cun 3949  cres 5687   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  selvvvval  42595
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