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Theorem offveqb 5672
 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 (φA 𝑉)
offveq.2 (φ𝐹 Fn A)
offveq.3 (φ𝐺 Fn A)
offveq.4 (φ𝐻 Fn A)
offveq.5 ((φ x A) → (𝐹x) = B)
offveq.6 ((φ x A) → (𝐺x) = 𝐶)
Assertion
Ref Expression
offveqb (φ → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ x A (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 5162 . . . 4 (𝐻 Fn A𝐻 = (x A ↦ (𝐻x)))
31, 2syl 14 . . 3 (φ𝐻 = (x A ↦ (𝐻x)))
4 offveq.2 . . . 4 (φ𝐹 Fn A)
5 offveq.3 . . . 4 (φ𝐺 Fn A)
6 offveq.1 . . . 4 (φA 𝑉)
7 inidm 3140 . . . 4 (AA) = A
8 offveq.5 . . . 4 ((φ x A) → (𝐹x) = B)
9 offveq.6 . . . 4 ((φ x A) → (𝐺x) = 𝐶)
104, 5, 6, 6, 7, 8, 9offval 5661 . . 3 (φ → (𝐹𝑓 𝑅𝐺) = (x A ↦ (B𝑅𝐶)))
113, 10eqeq12d 2051 . 2 (φ → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ (x A ↦ (𝐻x)) = (x A ↦ (B𝑅𝐶))))
12 funfvex 5135 . . . . . 6 ((Fun 𝐻 x dom 𝐻) → (𝐻x) V)
1312funfni 4942 . . . . 5 ((𝐻 Fn A x A) → (𝐻x) V)
141, 13sylan 267 . . . 4 ((φ x A) → (𝐻x) V)
1514ralrimiva 2386 . . 3 (φx A (𝐻x) V)
16 mpteqb 5204 . . 3 (x A (𝐻x) V → ((x A ↦ (𝐻x)) = (x A ↦ (B𝑅𝐶)) ↔ x A (𝐻x) = (B𝑅𝐶)))
1715, 16syl 14 . 2 (φ → ((x A ↦ (𝐻x)) = (x A ↦ (B𝑅𝐶)) ↔ x A (𝐻x) = (B𝑅𝐶)))
1811, 17bitrd 177 1 (φ → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ x A (𝐻x) = (B𝑅𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ↦ cmpt 3809   Fn wfn 4840  ‘cfv 4845  (class class class)co 5455   ∘𝑓 cof 5652 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654 This theorem is referenced by: (None)
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