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Theorem offveqb 6961
 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 6280 . . . 4 (𝐻 Fn 𝐴𝐻 = (𝑥𝐴 ↦ (𝐻𝑥)))
31, 2sylib 208 . . 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 3855 . . . 4 (𝐴𝐴) = 𝐴
8 offveq.5 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
9 offveq.6 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
104, 5, 6, 6, 7, 8, 9offval 6946 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
113, 10eqeq12d 2666 . 2 (𝜑 → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ (𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))))
12 fvexd 6241 . . . 4 (𝜑 → (𝐻𝑥) ∈ V)
1312ralrimivw 2996 . . 3 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) ∈ V)
14 mpteqb 6338 . . 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 268 1 (𝜑 → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ↦ cmpt 4762   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939 This theorem is referenced by:  eqlkr2  34705
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