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Theorem off2 29954
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 473 . . . . 5 ((𝜑𝑧𝐶) → 𝐹:𝐴𝑆)
3 off2.6 . . . . . . 7 (𝜑 → (𝐴𝐵) = 𝐶)
4 inss1 4026 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
53, 4syl6eqssr 3850 . . . . . 6 (𝜑𝐶𝐴)
65sselda 3796 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐴)
72, 6ffvelrnd 6584 . . . 4 ((𝜑𝑧𝐶) → (𝐹𝑧) ∈ 𝑆)
8 off2.3 . . . . . 6 (𝜑𝐺:𝐵𝑇)
98adantr 473 . . . . 5 ((𝜑𝑧𝐶) → 𝐺:𝐵𝑇)
10 inss2 4027 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
113, 10syl6eqssr 3850 . . . . . 6 (𝜑𝐶𝐵)
1211sselda 3796 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐵)
139, 12ffvelrnd 6584 . . . 4 ((𝜑𝑧𝐶) → (𝐺𝑧) ∈ 𝑇)
14 off2.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
1514ralrimivva 3150 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1615adantr 473 . . . 4 ((𝜑𝑧𝐶) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17 ovrspc2v 6902 . . . 4 ((((𝐹𝑧) ∈ 𝑆 ∧ (𝐺𝑧) ∈ 𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
187, 13, 16, 17syl21anc 867 . . 3 ((𝜑𝑧𝐶) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
1918fmpttd 6609 . 2 (𝜑 → (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈)
201ffnd 6255 . . . . 5 (𝜑𝐹 Fn 𝐴)
218ffnd 6255 . . . . 5 (𝜑𝐺 Fn 𝐵)
22 off2.4 . . . . 5 (𝜑𝐴𝑉)
23 off2.5 . . . . 5 (𝜑𝐵𝑊)
24 eqid 2797 . . . . 5 (𝐴𝐵) = (𝐴𝐵)
25 eqidd 2798 . . . . 5 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
26 eqidd 2798 . . . . 5 ((𝜑𝑧𝐵) → (𝐺𝑧) = (𝐺𝑧))
2720, 21, 22, 23, 24, 25, 26offval 7136 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
283mpteq1d 4929 . . . 4 (𝜑 → (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
2927, 28eqtrd 2831 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3029feq1d 6239 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺):𝐶𝑈 ↔ (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈))
3119, 30mpbird 249 1 (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3087  cin 3766  cmpt 4920  wf 6095  cfv 6099  (class class class)co 6876  𝑓 cof 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-of 7129
This theorem is referenced by: (None)
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