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Theorem offveqb 7647
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 6906 . . . 4 (𝐻 Fn 𝐴𝐻 = (𝑥𝐴 ↦ (𝐻𝑥)))
31, 2sylib 217 . . 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 4183 . . . 4 (𝐴𝐴) = 𝐴
8 offveq.5 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
9 offveq.6 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
104, 5, 6, 6, 7, 8, 9offval 7631 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
113, 10eqeq12d 2753 . 2 (𝜑 → (𝐻 = (𝐹f 𝑅𝐺) ↔ (𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))))
12 fvexd 6862 . . . 4 (𝜑 → (𝐻𝑥) ∈ V)
1312ralrimivw 3148 . . 3 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) ∈ V)
14 mpteqb 6972 . . 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 205  wa 397   = wceq 1542  wcel 2107  wral 3065  Vcvv 3448  cmpt 5193   Fn wfn 6496  cfv 6501  (class class class)co 7362  f cof 7620
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  eqlkr2  37591
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