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Theorem off2 28823
 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 . . . . . 6 (𝜑𝐹:𝐴𝑆)
21adantr 480 . . . . 5 ((𝜑𝑧𝐶) → 𝐹:𝐴𝑆)
3 off2.6 . . . . . . 7 (𝜑 → (𝐴𝐵) = 𝐶)
4 inss1 3795 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
53, 4syl6eqssr 3619 . . . . . 6 (𝜑𝐶𝐴)
65sselda 3568 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐴)
72, 6ffvelrnd 6268 . . . 4 ((𝜑𝑧𝐶) → (𝐹𝑧) ∈ 𝑆)
8 off2.3 . . . . . 6 (𝜑𝐺:𝐵𝑇)
98adantr 480 . . . . 5 ((𝜑𝑧𝐶) → 𝐺:𝐵𝑇)
10 inss2 3796 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
113, 10syl6eqssr 3619 . . . . . 6 (𝜑𝐶𝐵)
1211sselda 3568 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐵)
139, 12ffvelrnd 6268 . . . 4 ((𝜑𝑧𝐶) → (𝐺𝑧) ∈ 𝑇)
14 off2.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
1514ralrimivva 2954 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1615adantr 480 . . . 4 ((𝜑𝑧𝐶) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17 ovrspc2v 6571 . . . 4 ((((𝐹𝑧) ∈ 𝑆 ∧ (𝐺𝑧) ∈ 𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
187, 13, 16, 17syl21anc 1317 . . 3 ((𝜑𝑧𝐶) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
19 eqid 2610 . . 3 (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧)))
2018, 19fmptd 6292 . 2 (𝜑 → (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈)
21 ffn 5958 . . . . . 6 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
221, 21syl 17 . . . . 5 (𝜑𝐹 Fn 𝐴)
23 ffn 5958 . . . . . 6 (𝐺:𝐵𝑇𝐺 Fn 𝐵)
248, 23syl 17 . . . . 5 (𝜑𝐺 Fn 𝐵)
25 off2.4 . . . . 5 (𝜑𝐴𝑉)
26 off2.5 . . . . 5 (𝜑𝐵𝑊)
27 eqid 2610 . . . . 5 (𝐴𝐵) = (𝐴𝐵)
28 eqidd 2611 . . . . 5 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
29 eqidd 2611 . . . . 5 ((𝜑𝑧𝐵) → (𝐺𝑧) = (𝐺𝑧))
3022, 24, 25, 26, 27, 28, 29offval 6802 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
313mpteq1d 4666 . . . 4 (𝜑 → (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3230, 31eqtrd 2644 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3332feq1d 5943 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺):𝐶𝑈 ↔ (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈))
3420, 33mpbird 246 1 (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ↦ cmpt 4643   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795 This theorem is referenced by: (None)
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