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Theorem off2 29777
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 466 . . . . 5 ((𝜑𝑧𝐶) → 𝐹:𝐴𝑆)
3 off2.6 . . . . . . 7 (𝜑 → (𝐴𝐵) = 𝐶)
4 inss1 3979 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
53, 4syl6eqssr 3803 . . . . . 6 (𝜑𝐶𝐴)
65sselda 3750 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐴)
72, 6ffvelrnd 6503 . . . 4 ((𝜑𝑧𝐶) → (𝐹𝑧) ∈ 𝑆)
8 off2.3 . . . . . 6 (𝜑𝐺:𝐵𝑇)
98adantr 466 . . . . 5 ((𝜑𝑧𝐶) → 𝐺:𝐵𝑇)
10 inss2 3980 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
113, 10syl6eqssr 3803 . . . . . 6 (𝜑𝐶𝐵)
1211sselda 3750 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐵)
139, 12ffvelrnd 6503 . . . 4 ((𝜑𝑧𝐶) → (𝐺𝑧) ∈ 𝑇)
14 off2.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
1514ralrimivva 3119 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1615adantr 466 . . . 4 ((𝜑𝑧𝐶) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17 ovrspc2v 6816 . . . 4 ((((𝐹𝑧) ∈ 𝑆 ∧ (𝐺𝑧) ∈ 𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
187, 13, 16, 17syl21anc 1474 . . 3 ((𝜑𝑧𝐶) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
19 eqid 2770 . . 3 (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧)))
2018, 19fmptd 6527 . 2 (𝜑 → (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈)
21 ffn 6185 . . . . . 6 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
221, 21syl 17 . . . . 5 (𝜑𝐹 Fn 𝐴)
23 ffn 6185 . . . . . 6 (𝐺:𝐵𝑇𝐺 Fn 𝐵)
248, 23syl 17 . . . . 5 (𝜑𝐺 Fn 𝐵)
25 off2.4 . . . . 5 (𝜑𝐴𝑉)
26 off2.5 . . . . 5 (𝜑𝐵𝑊)
27 eqid 2770 . . . . 5 (𝐴𝐵) = (𝐴𝐵)
28 eqidd 2771 . . . . 5 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
29 eqidd 2771 . . . . 5 ((𝜑𝑧𝐵) → (𝐺𝑧) = (𝐺𝑧))
3022, 24, 25, 26, 27, 28, 29offval 7050 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
313mpteq1d 4870 . . . 4 (𝜑 → (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3230, 31eqtrd 2804 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3332feq1d 6170 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺):𝐶𝑈 ↔ (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈))
3420, 33mpbird 247 1 (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  cin 3720  cmpt 4861   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6792  𝑓 cof 7041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043
This theorem is referenced by: (None)
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