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Theorem offveqb 6001
 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 dffn5im 5467 . . . 4 (𝐻 Fn 𝐴𝐻 = (𝑥𝐴 ↦ (𝐻𝑥)))
31, 2syl 14 . . 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 3285 . . . 4 (𝐴𝐴) = 𝐴
8 offveq.5 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
9 offveq.6 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)
104, 5, 6, 6, 7, 8, 9offval 5989 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
113, 10eqeq12d 2154 . 2 (𝜑 → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ (𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))))
12 funfvex 5438 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → (𝐻𝑥) ∈ V)
1312funfni 5223 . . . . 5 ((𝐻 Fn 𝐴𝑥𝐴) → (𝐻𝑥) ∈ V)
141, 13sylan 281 . . . 4 ((𝜑𝑥𝐴) → (𝐻𝑥) ∈ V)
1514ralrimiva 2505 . . 3 (𝜑 → ∀𝑥𝐴 (𝐻𝑥) ∈ V)
16 mpteqb 5511 . . 3 (∀𝑥𝐴 (𝐻𝑥) ∈ V → ((𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
1715, 16syl 14 . 2 (𝜑 → ((𝑥𝐴 ↦ (𝐻𝑥)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
1811, 17bitrd 187 1 (𝜑 → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   ↦ cmpt 3989   Fn wfn 5118  ‘cfv 5123  (class class class)co 5774   ∘𝑓 cof 5980 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982 This theorem is referenced by: (None)
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