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Theorem off2 30374
 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 6488 . . . 4 (𝜑𝐹 Fn 𝐴)
3 off2.3 . . . . 5 (𝜑𝐺:𝐵𝑇)
43ffnd 6488 . . . 4 (𝜑𝐺 Fn 𝐵)
5 off2.4 . . . 4 (𝜑𝐴𝑉)
6 off2.5 . . . 4 (𝜑𝐵𝑊)
7 eqid 2821 . . . 4 (𝐴𝐵) = (𝐴𝐵)
8 eqidd 2822 . . . 4 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
9 eqidd 2822 . . . 4 ((𝜑𝑧𝐵) → (𝐺𝑧) = (𝐺𝑧))
102, 4, 5, 6, 7, 8, 9offval 7391 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
11 off2.6 . . . 4 (𝜑 → (𝐴𝐵) = 𝐶)
1211mpteq1d 5128 . . 3 (𝜑 → (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
1310, 12eqtrd 2856 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
141adantr 484 . . . 4 ((𝜑𝑧𝐶) → 𝐹:𝐴𝑆)
15 inss1 4180 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
1611, 15eqsstrrdi 3998 . . . . 5 (𝜑𝐶𝐴)
1716sselda 3943 . . . 4 ((𝜑𝑧𝐶) → 𝑧𝐴)
1814, 17ffvelrnd 6825 . . 3 ((𝜑𝑧𝐶) → (𝐹𝑧) ∈ 𝑆)
193adantr 484 . . . 4 ((𝜑𝑧𝐶) → 𝐺:𝐵𝑇)
20 inss2 4181 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
2111, 20eqsstrrdi 3998 . . . . 5 (𝜑𝐶𝐵)
2221sselda 3943 . . . 4 ((𝜑𝑧𝐶) → 𝑧𝐵)
2319, 22ffvelrnd 6825 . . 3 ((𝜑𝑧𝐶) → (𝐺𝑧) ∈ 𝑇)
24 off2.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
2524ralrimivva 3179 . . . 4 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
2625adantr 484 . . 3 ((𝜑𝑧𝐶) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
27 ovrspc2v 7156 . . 3 ((((𝐹𝑧) ∈ 𝑆 ∧ (𝐺𝑧) ∈ 𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
2818, 23, 26, 27syl21anc 836 . 2 ((𝜑𝑧𝐶) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
2913, 28fmpt3d 6853 1 (𝜑 → (𝐹f 𝑅𝐺):𝐶𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3126   ∩ cin 3909   ↦ cmpt 5119  ⟶wf 6324  ‘cfv 6328  (class class class)co 7130   ∘f cof 7382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384 This theorem is referenced by: (None)
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