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Theorem off 6109
Description: The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
off.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
off.2 (𝜑𝐹:𝐴𝑆)
off.3 (𝜑𝐺:𝐵𝑇)
off.4 (𝜑𝐴𝑉)
off.5 (𝜑𝐵𝑊)
off.6 (𝐴𝐵) = 𝐶
Assertion
Ref Expression
off (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
Distinct variable groups:   𝑦,𝐺   𝑥,𝑦,𝜑   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem off
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 off.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
2 off.6 . . . . . . 7 (𝐴𝐵) = 𝐶
3 inss1 3367 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
42, 3eqsstrri 3200 . . . . . 6 𝐶𝐴
54sseli 3163 . . . . 5 (𝑧𝐶𝑧𝐴)
6 ffvelcdm 5662 . . . . 5 ((𝐹:𝐴𝑆𝑧𝐴) → (𝐹𝑧) ∈ 𝑆)
71, 5, 6syl2an 289 . . . 4 ((𝜑𝑧𝐶) → (𝐹𝑧) ∈ 𝑆)
8 off.3 . . . . 5 (𝜑𝐺:𝐵𝑇)
9 inss2 3368 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
102, 9eqsstrri 3200 . . . . . 6 𝐶𝐵
1110sseli 3163 . . . . 5 (𝑧𝐶𝑧𝐵)
12 ffvelcdm 5662 . . . . 5 ((𝐺:𝐵𝑇𝑧𝐵) → (𝐺𝑧) ∈ 𝑇)
138, 11, 12syl2an 289 . . . 4 ((𝜑𝑧𝐶) → (𝐺𝑧) ∈ 𝑇)
14 off.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
1514ralrimivva 2569 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1615adantr 276 . . . 4 ((𝜑𝑧𝐶) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17 oveq1 5895 . . . . . 6 (𝑥 = (𝐹𝑧) → (𝑥𝑅𝑦) = ((𝐹𝑧)𝑅𝑦))
1817eleq1d 2256 . . . . 5 (𝑥 = (𝐹𝑧) → ((𝑥𝑅𝑦) ∈ 𝑈 ↔ ((𝐹𝑧)𝑅𝑦) ∈ 𝑈))
19 oveq2 5896 . . . . . 6 (𝑦 = (𝐺𝑧) → ((𝐹𝑧)𝑅𝑦) = ((𝐹𝑧)𝑅(𝐺𝑧)))
2019eleq1d 2256 . . . . 5 (𝑦 = (𝐺𝑧) → (((𝐹𝑧)𝑅𝑦) ∈ 𝑈 ↔ ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈))
2118, 20rspc2va 2867 . . . 4 ((((𝐹𝑧) ∈ 𝑆 ∧ (𝐺𝑧) ∈ 𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
227, 13, 16, 21syl21anc 1247 . . 3 ((𝜑𝑧𝐶) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
23 eqid 2187 . . 3 (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧)))
2422, 23fmptd 5683 . 2 (𝜑 → (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈)
25 ffn 5377 . . . . 5 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
261, 25syl 14 . . . 4 (𝜑𝐹 Fn 𝐴)
27 ffn 5377 . . . . 5 (𝐺:𝐵𝑇𝐺 Fn 𝐵)
288, 27syl 14 . . . 4 (𝜑𝐺 Fn 𝐵)
29 off.4 . . . 4 (𝜑𝐴𝑉)
30 off.5 . . . 4 (𝜑𝐵𝑊)
31 eqidd 2188 . . . 4 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
32 eqidd 2188 . . . 4 ((𝜑𝑧𝐵) → (𝐺𝑧) = (𝐺𝑧))
3326, 28, 29, 30, 2, 31, 32offval 6104 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3433feq1d 5364 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺):𝐶𝑈 ↔ (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈))
3524, 34mpbird 167 1 (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  wral 2465  cin 3140  cmpt 4076   Fn wfn 5223  wf 5224  cfv 5228  (class class class)co 5888  𝑓 cof 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-of 6097
This theorem is referenced by:  offeq  6110  lcomf  13573  dvaddxxbr  14518  dvmulxxbr  14519  dvaddxx  14520  dvmulxx  14521  dviaddf  14522  dvimulf  14523
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