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Theorem offveqb 5649
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 5140 . . . 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 3119 . . . 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 5638 . . 3 (φ → (𝐹𝑓 𝑅𝐺) = (x A ↦ (B𝑅𝐶)))
113, 10eqeq12d 2032 . 2 (φ → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ (x A ↦ (𝐻x)) = (x A ↦ (B𝑅𝐶))))
12 funfvex 5113 . . . . . 6 ((Fun 𝐻 x dom 𝐻) → (𝐻x) V)
1312funfni 4921 . . . . 5 ((𝐻 Fn A x A) → (𝐻x) V)
141, 13sylan 267 . . . 4 ((φ x A) → (𝐻x) V)
1514ralrimiva 2366 . . 3 (φx A (𝐻x) V)
16 mpteqb 5182 . . 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 1226   wcel 1370  wral 2280  Vcvv 2531  cmpt 3788   Fn wfn 4820  cfv 4825  (class class class)co 5432  𝑓 cof 5629
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-of 5631
This theorem is referenced by: (None)
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