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Theorem offval 5630
Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1 (φ𝐹 Fn A)
offval.2 (φ𝐺 Fn B)
offval.3 (φA 𝑉)
offval.4 (φB 𝑊)
offval.5 (AB) = 𝑆
offval.6 ((φ x A) → (𝐹x) = 𝐶)
offval.7 ((φ x B) → (𝐺x) = 𝐷)
Assertion
Ref Expression
offval (φ → (𝐹𝑓 𝑅𝐺) = (x 𝑆 ↦ (𝐶𝑅𝐷)))
Distinct variable groups:   x,A   x,𝐹   x,𝐺   φ,x   x,𝑆   x,𝑅
Allowed substitution hints:   B(x)   𝐶(x)   𝐷(x)   𝑉(x)   𝑊(x)

Proof of Theorem offval
Dummy variables f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4 (φ𝐹 Fn A)
2 offval.3 . . . 4 (φA 𝑉)
3 fnex 5296 . . . 4 ((𝐹 Fn A A 𝑉) → 𝐹 V)
41, 2, 3syl2anc 391 . . 3 (φ𝐹 V)
5 offval.2 . . . 4 (φ𝐺 Fn B)
6 offval.4 . . . 4 (φB 𝑊)
7 fnex 5296 . . . 4 ((𝐺 Fn B B 𝑊) → 𝐺 V)
85, 6, 7syl2anc 391 . . 3 (φ𝐺 V)
9 fndm 4912 . . . . . . . 8 (𝐹 Fn A → dom 𝐹 = A)
101, 9syl 14 . . . . . . 7 (φ → dom 𝐹 = A)
11 fndm 4912 . . . . . . . 8 (𝐺 Fn B → dom 𝐺 = B)
125, 11syl 14 . . . . . . 7 (φ → dom 𝐺 = B)
1310, 12ineq12d 3107 . . . . . 6 (φ → (dom 𝐹 ∩ dom 𝐺) = (AB))
14 offval.5 . . . . . 6 (AB) = 𝑆
1513, 14syl6eq 2061 . . . . 5 (φ → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
1615mpteq1d 3805 . . . 4 (φ → (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))) = (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))))
17 inex1g 3856 . . . . . 6 (A 𝑉 → (AB) V)
1814, 17syl5eqelr 2098 . . . . 5 (A 𝑉𝑆 V)
19 mptexg 5299 . . . . 5 (𝑆 V → (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) V)
202, 18, 193syl 17 . . . 4 (φ → (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) V)
2116, 20eqeltrd 2087 . . 3 (φ → (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))) V)
22 dmeq 4450 . . . . . 6 (f = 𝐹 → dom f = dom 𝐹)
23 dmeq 4450 . . . . . 6 (g = 𝐺 → dom g = dom 𝐺)
2422, 23ineqan12d 3108 . . . . 5 ((f = 𝐹 g = 𝐺) → (dom f ∩ dom g) = (dom 𝐹 ∩ dom 𝐺))
25 fveq1 5090 . . . . . 6 (f = 𝐹 → (fx) = (𝐹x))
26 fveq1 5090 . . . . . 6 (g = 𝐺 → (gx) = (𝐺x))
2725, 26oveqan12d 5443 . . . . 5 ((f = 𝐹 g = 𝐺) → ((fx)𝑅(gx)) = ((𝐹x)𝑅(𝐺x)))
2824, 27mpteq12dv 3802 . . . 4 ((f = 𝐹 g = 𝐺) → (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))) = (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))))
29 df-of 5623 . . . 4 𝑓 𝑅 = (f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
3028, 29ovmpt2ga 5541 . . 3 ((𝐹 V 𝐺 V (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))) V) → (𝐹𝑓 𝑅𝐺) = (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))))
314, 8, 21, 30syl3anc 1116 . 2 (φ → (𝐹𝑓 𝑅𝐺) = (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))))
3214eleq2i 2077 . . . . 5 (x (AB) ↔ x 𝑆)
33 elin 3094 . . . . 5 (x (AB) ↔ (x A x B))
3432, 33bitr3i 175 . . . 4 (x 𝑆 ↔ (x A x B))
35 offval.6 . . . . . 6 ((φ x A) → (𝐹x) = 𝐶)
3635adantrr 448 . . . . 5 ((φ (x A x B)) → (𝐹x) = 𝐶)
37 offval.7 . . . . . 6 ((φ x B) → (𝐺x) = 𝐷)
3837adantrl 447 . . . . 5 ((φ (x A x B)) → (𝐺x) = 𝐷)
3936, 38oveq12d 5442 . . . 4 ((φ (x A x B)) → ((𝐹x)𝑅(𝐺x)) = (𝐶𝑅𝐷))
4034, 39sylan2b 271 . . 3 ((φ x 𝑆) → ((𝐹x)𝑅(𝐺x)) = (𝐶𝑅𝐷))
4140mpteq2dva 3810 . 2 (φ → (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) = (x 𝑆 ↦ (𝐶𝑅𝐷)))
4231, 16, 413eqtrd 2049 1 (φ → (𝐹𝑓 𝑅𝐺) = (x 𝑆 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223   wcel 1366  Vcvv 2526  cin 2884  cmpt 3781  dom cdm 4260   Fn wfn 4812  cfv 4817  (class class class)co 5424  𝑓 cof 5621
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-pow 3890  ax-pr 3907  ax-setind 4192
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ne 2179  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-ov 5427  df-oprab 5428  df-mpt2 5429  df-of 5623
This theorem is referenced by:  fnofval  5632  off  5635  ofres  5636  offval2  5637  suppssof1  5639  ofco  5640  offveqb  5641
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