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Theorem offveqb 7420
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 6717 . . . 4 (𝐻 Fn 𝐴𝐻 = (𝑥𝐴 ↦ (𝐻𝑥)))
31, 2sylib 219 . . 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 4192 . . . 4 (𝐴𝐴) = 𝐴
8 offveq.5 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
9 offveq.6 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
104, 5, 6, 6, 7, 8, 9offval 7405 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
113, 10eqeq12d 2834 . 2 (𝜑 → (𝐻 = (𝐹f 𝑅𝐺) ↔ (𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))))
12 fvexd 6678 . . . 4 (𝜑 → (𝐻𝑥) ∈ V)
1312ralrimivw 3180 . . 3 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) ∈ V)
14 mpteqb 6779 . . 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 280 1 (𝜑 → (𝐻 = (𝐹f 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cmpt 5137   Fn wfn 6343  cfv 6348  (class class class)co 7145  f cof 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398
This theorem is referenced by:  eqlkr2  36116
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