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Theorem off2 32651
Description: The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
Hypotheses
Ref Expression
off2.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
off2.2 (𝜑𝐹:𝐴𝑆)
off2.3 (𝜑𝐺:𝐵𝑇)
off2.4 (𝜑𝐴𝑉)
off2.5 (𝜑𝐵𝑊)
off2.6 (𝜑 → (𝐴𝐵) = 𝐶)
Assertion
Ref Expression
off2 (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 off2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 off2.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
21ffnd 6737 . . . 4 (𝜑𝐹 Fn 𝐴)
3 off2.3 . . . . 5 (𝜑𝐺:𝐵𝑇)
43ffnd 6737 . . . 4 (𝜑𝐺 Fn 𝐵)
5 off2.4 . . . 4 (𝜑𝐴𝑉)
6 off2.5 . . . 4 (𝜑𝐵𝑊)
7 eqid 2737 . . . 4 (𝐴𝐵) = (𝐴𝐵)
8 eqidd 2738 . . . 4 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
9 eqidd 2738 . . . 4 ((𝜑𝑧𝐵) → (𝐺𝑧) = (𝐺𝑧))
102, 4, 5, 6, 7, 8, 9offval 7706 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
11 off2.6 . . . 4 (𝜑 → (𝐴𝐵) = 𝐶)
1211mpteq1d 5237 . . 3 (𝜑 → (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
1310, 12eqtrd 2777 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
141adantr 480 . . . 4 ((𝜑𝑧𝐶) → 𝐹:𝐴𝑆)
15 inss1 4237 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
1611, 15eqsstrrdi 4029 . . . . 5 (𝜑𝐶𝐴)
1716sselda 3983 . . . 4 ((𝜑𝑧𝐶) → 𝑧𝐴)
1814, 17ffvelcdmd 7105 . . 3 ((𝜑𝑧𝐶) → (𝐹𝑧) ∈ 𝑆)
193adantr 480 . . . 4 ((𝜑𝑧𝐶) → 𝐺:𝐵𝑇)
20 inss2 4238 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
2111, 20eqsstrrdi 4029 . . . . 5 (𝜑𝐶𝐵)
2221sselda 3983 . . . 4 ((𝜑𝑧𝐶) → 𝑧𝐵)
2319, 22ffvelcdmd 7105 . . 3 ((𝜑𝑧𝐶) → (𝐺𝑧) ∈ 𝑇)
24 off2.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
2524ralrimivva 3202 . . . 4 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
2625adantr 480 . . 3 ((𝜑𝑧𝐶) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
27 ovrspc2v 7457 . . 3 ((((𝐹𝑧) ∈ 𝑆 ∧ (𝐺𝑧) ∈ 𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
2818, 23, 26, 27syl21anc 838 . 2 ((𝜑𝑧𝐶) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
2913, 28fmpt3d 7136 1 (𝜑 → (𝐹f 𝑅𝐺):𝐶𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  cin 3950  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by: (None)
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