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Theorem offveqb 7724
Description: Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
offveq.1 (𝜑𝐴𝑉)
offveq.2 (𝜑𝐹 Fn 𝐴)
offveq.3 (𝜑𝐺 Fn 𝐴)
offveq.4 (𝜑𝐻 Fn 𝐴)
offveq.5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
offveq.6 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
Assertion
Ref Expression
offveqb (𝜑 → (𝐻 = (𝐹f 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻   𝜑,𝑥   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem offveqb
StepHypRef Expression
1 offveq.4 . . . 4 (𝜑𝐻 Fn 𝐴)
2 dffn5 6967 . . . 4 (𝐻 Fn 𝐴𝐻 = (𝑥𝐴 ↦ (𝐻𝑥)))
31, 2sylib 218 . . 3 (𝜑𝐻 = (𝑥𝐴 ↦ (𝐻𝑥)))
4 offveq.2 . . . 4 (𝜑𝐹 Fn 𝐴)
5 offveq.3 . . . 4 (𝜑𝐺 Fn 𝐴)
6 offveq.1 . . . 4 (𝜑𝐴𝑉)
7 inidm 4235 . . . 4 (𝐴𝐴) = 𝐴
8 offveq.5 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
9 offveq.6 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
104, 5, 6, 6, 7, 8, 9offval 7706 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
113, 10eqeq12d 2751 . 2 (𝜑 → (𝐻 = (𝐹f 𝑅𝐺) ↔ (𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))))
12 fvexd 6922 . . . 4 (𝜑 → (𝐻𝑥) ∈ V)
1312ralrimivw 3148 . . 3 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) ∈ V)
14 mpteqb 7035 . . 3 (∀𝑥𝐴 (𝐻𝑥) ∈ V → ((𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
1513, 14syl 17 . 2 (𝜑 → ((𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
1611, 15bitrd 279 1 (𝜑 → (𝐻 = (𝐹f 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cmpt 5231   Fn wfn 6558  cfv 6563  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  eqlkr2  39082
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