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Theorem eqfnun 36231
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 5935 . . 3 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
2 reseq1 5935 . . 3 (𝐹 = 𝐺 → (𝐹𝐵) = (𝐺𝐵))
31, 2jca 513 . 2 (𝐹 = 𝐺 → ((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)))
4 elun 4112 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 fveq1 6845 . . . . . . . . 9 ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐴)‘𝑥) = ((𝐺𝐴)‘𝑥))
6 fvres 6865 . . . . . . . . 9 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
75, 6sylan9req 2794 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐹𝑥))
8 fvres 6865 . . . . . . . . 9 (𝑥𝐴 → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
98adantl 483 . . . . . . . 8 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → ((𝐺𝐴)‘𝑥) = (𝐺𝑥))
107, 9eqtr3d 2775 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
1110adantlr 714 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
12 fveq1 6845 . . . . . . . . 9 ((𝐹𝐵) = (𝐺𝐵) → ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥))
13 fvres 6865 . . . . . . . . 9 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
1412, 13sylan9req 2794 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐹𝑥))
15 fvres 6865 . . . . . . . . 9 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1615adantl 483 . . . . . . . 8 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
1714, 16eqtr3d 2775 . . . . . . 7 (((𝐹𝐵) = (𝐺𝐵) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1817adantll 713 . . . . . 6 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
1911, 18jaodan 957 . . . . 5 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = (𝐺𝑥))
204, 19sylan2b 595 . . . 4 ((((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝐹𝑥) = (𝐺𝑥))
2120ralrimiva 3140 . . 3 (((𝐹𝐴) = (𝐺𝐴) ∧ (𝐹𝐵) = (𝐺𝐵)) → ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥))
22 eqfnfv 6986 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ 𝐺 Fn (𝐴𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = (𝐺𝑥)))
2321, 22syl5ibr 246 . 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 397  wo 846   = wceq 1542  wcel 2107  wral 3061  cun 3912  cres 5639   Fn wfn 6495  cfv 6500
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508
This theorem is referenced by: (None)
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