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Theorem offval2 5668
Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval2.1 (φA 𝑉)
offval2.2 ((φ x A) → B 𝑊)
offval2.3 ((φ x A) → 𝐶 𝑋)
offval2.4 (φ𝐹 = (x AB))
offval2.5 (φ𝐺 = (x A𝐶))
Assertion
Ref Expression
offval2 (φ → (𝐹𝑓 𝑅𝐺) = (x A ↦ (B𝑅𝐶)))
Distinct variable groups:   x,A   φ,x   x,𝑅
Allowed substitution hints:   B(x)   𝐶(x)   𝐹(x)   𝐺(x)   𝑉(x)   𝑊(x)   𝑋(x)

Proof of Theorem offval2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 offval2.2 . . . . . 6 ((φ x A) → B 𝑊)
21ralrimiva 2386 . . . . 5 (φx A B 𝑊)
3 eqid 2037 . . . . . 6 (x AB) = (x AB)
43fnmpt 4968 . . . . 5 (x A B 𝑊 → (x AB) Fn A)
52, 4syl 14 . . . 4 (φ → (x AB) Fn A)
6 offval2.4 . . . . 5 (φ𝐹 = (x AB))
76fneq1d 4932 . . . 4 (φ → (𝐹 Fn A ↔ (x AB) Fn A))
85, 7mpbird 156 . . 3 (φ𝐹 Fn A)
9 offval2.3 . . . . . 6 ((φ x A) → 𝐶 𝑋)
109ralrimiva 2386 . . . . 5 (φx A 𝐶 𝑋)
11 eqid 2037 . . . . . 6 (x A𝐶) = (x A𝐶)
1211fnmpt 4968 . . . . 5 (x A 𝐶 𝑋 → (x A𝐶) Fn A)
1310, 12syl 14 . . . 4 (φ → (x A𝐶) Fn A)
14 offval2.5 . . . . 5 (φ𝐺 = (x A𝐶))
1514fneq1d 4932 . . . 4 (φ → (𝐺 Fn A ↔ (x A𝐶) Fn A))
1613, 15mpbird 156 . . 3 (φ𝐺 Fn A)
17 offval2.1 . . 3 (φA 𝑉)
18 inidm 3140 . . 3 (AA) = A
196adantr 261 . . . 4 ((φ y A) → 𝐹 = (x AB))
2019fveq1d 5123 . . 3 ((φ y A) → (𝐹y) = ((x AB)‘y))
2114adantr 261 . . . 4 ((φ y A) → 𝐺 = (x A𝐶))
2221fveq1d 5123 . . 3 ((φ y A) → (𝐺y) = ((x A𝐶)‘y))
238, 16, 17, 17, 18, 20, 22offval 5661 . 2 (φ → (𝐹𝑓 𝑅𝐺) = (y A ↦ (((x AB)‘y)𝑅((x A𝐶)‘y))))
24 nffvmpt1 5129 . . . . 5 x((x AB)‘y)
25 nfcv 2175 . . . . 5 x𝑅
26 nffvmpt1 5129 . . . . 5 x((x A𝐶)‘y)
2724, 25, 26nfov 5478 . . . 4 x(((x AB)‘y)𝑅((x A𝐶)‘y))
28 nfcv 2175 . . . 4 y(((x AB)‘x)𝑅((x A𝐶)‘x))
29 fveq2 5121 . . . . 5 (y = x → ((x AB)‘y) = ((x AB)‘x))
30 fveq2 5121 . . . . 5 (y = x → ((x A𝐶)‘y) = ((x A𝐶)‘x))
3129, 30oveq12d 5473 . . . 4 (y = x → (((x AB)‘y)𝑅((x A𝐶)‘y)) = (((x AB)‘x)𝑅((x A𝐶)‘x)))
3227, 28, 31cbvmpt 3842 . . 3 (y A ↦ (((x AB)‘y)𝑅((x A𝐶)‘y))) = (x A ↦ (((x AB)‘x)𝑅((x A𝐶)‘x)))
33 simpr 103 . . . . . 6 ((φ x A) → x A)
343fvmpt2 5197 . . . . . 6 ((x A B 𝑊) → ((x AB)‘x) = B)
3533, 1, 34syl2anc 391 . . . . 5 ((φ x A) → ((x AB)‘x) = B)
3611fvmpt2 5197 . . . . . 6 ((x A 𝐶 𝑋) → ((x A𝐶)‘x) = 𝐶)
3733, 9, 36syl2anc 391 . . . . 5 ((φ x A) → ((x A𝐶)‘x) = 𝐶)
3835, 37oveq12d 5473 . . . 4 ((φ x A) → (((x AB)‘x)𝑅((x A𝐶)‘x)) = (B𝑅𝐶))
3938mpteq2dva 3838 . . 3 (φ → (x A ↦ (((x AB)‘x)𝑅((x A𝐶)‘x))) = (x A ↦ (B𝑅𝐶)))
4032, 39syl5eq 2081 . 2 (φ → (y A ↦ (((x AB)‘y)𝑅((x A𝐶)‘y))) = (x A ↦ (B𝑅𝐶)))
4123, 40eqtrd 2069 1 (φ → (𝐹𝑓 𝑅𝐺) = (x A ↦ (B𝑅𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wral 2300  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-bnd 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:  ofc12  5673  caofinvl  5675  caofcom  5676
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