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Theorem eqfnun 33848
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 5528 . . 3 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
2 reseq1 5528 . . 3 (𝐹 = 𝐺 → (𝐹𝐵) = (𝐺𝐵))
31, 2jca 501 . 2 (𝐹 = 𝐺 → ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)))
4 elun 3904 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 fveq1 6331 . . . . . . . . 9 ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐴)‘𝑥) = ((𝐺𝐴)‘𝑥))
6 fvres 6348 . . . . . . . . 9 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
75, 6sylan9req 2826 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐹𝑥))
8 fvres 6348 . . . . . . . . 9 (𝑥𝐴 → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
98adantl 467 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
107, 9eqtr3d 2807 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
1110adantlr 694 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
12 fveq1 6331 . . . . . . . . 9 ((𝐹𝐵) = (𝐺𝐵) → ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥))
13 fvres 6348 . . . . . . . . 9 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
1412, 13sylan9req 2826 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐹𝑥))
15 fvres 6348 . . . . . . . . 9 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1615adantl 467 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1714, 16eqtr3d 2807 . . . . . . 7 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1817adantll 693 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1911, 18jaodan 942 . . . . 5 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = (𝐺𝑥))
204, 19sylan2b 581 . . . 4 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝐹𝑥) = (𝐺𝑥))
2120ralrimiva 3115 . . 3 (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥))
22 eqfnfv 6454 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥)))
2321, 22syl5ibr 236 . 2 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → 𝐹 = 𝐺))
243, 23impbid2 216 1 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836   = wceq 1631  wcel 2145  wral 3061  cun 3721  cres 5251   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by: (None)
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