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Theorem off2 29308
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 481 . . . . 5 ((𝜑𝑧𝐶) → 𝐹:𝐴𝑆)
3 off2.6 . . . . . . 7 (𝜑 → (𝐴𝐵) = 𝐶)
4 inss1 3816 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
53, 4syl6eqssr 3640 . . . . . 6 (𝜑𝐶𝐴)
65sselda 3587 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐴)
72, 6ffvelrnd 6321 . . . 4 ((𝜑𝑧𝐶) → (𝐹𝑧) ∈ 𝑆)
8 off2.3 . . . . . 6 (𝜑𝐺:𝐵𝑇)
98adantr 481 . . . . 5 ((𝜑𝑧𝐶) → 𝐺:𝐵𝑇)
10 inss2 3817 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
113, 10syl6eqssr 3640 . . . . . 6 (𝜑𝐶𝐵)
1211sselda 3587 . . . . 5 ((𝜑𝑧𝐶) → 𝑧𝐵)
139, 12ffvelrnd 6321 . . . 4 ((𝜑𝑧𝐶) → (𝐺𝑧) ∈ 𝑇)
14 off2.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
1514ralrimivva 2966 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1615adantr 481 . . . 4 ((𝜑𝑧𝐶) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17 ovrspc2v 6632 . . . 4 ((((𝐹𝑧) ∈ 𝑆 ∧ (𝐺𝑧) ∈ 𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
187, 13, 16, 17syl21anc 1322 . . 3 ((𝜑𝑧𝐶) → ((𝐹𝑧)𝑅(𝐺𝑧)) ∈ 𝑈)
19 eqid 2621 . . 3 (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧)))
2018, 19fmptd 6346 . 2 (𝜑 → (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))):𝐶𝑈)
21 ffn 6007 . . . . . 6 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
221, 21syl 17 . . . . 5 (𝜑𝐹 Fn 𝐴)
23 ffn 6007 . . . . . 6 (𝐺:𝐵𝑇𝐺 Fn 𝐵)
248, 23syl 17 . . . . 5 (𝜑𝐺 Fn 𝐵)
25 off2.4 . . . . 5 (𝜑𝐴𝑉)
26 off2.5 . . . . 5 (𝜑𝐵𝑊)
27 eqid 2621 . . . . 5 (𝐴𝐵) = (𝐴𝐵)
28 eqidd 2622 . . . . 5 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
29 eqidd 2622 . . . . 5 ((𝜑𝑧𝐵) → (𝐺𝑧) = (𝐺𝑧))
3022, 24, 25, 26, 27, 28, 29offval 6864 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
313mpteq1d 4703 . . . 4 (𝜑 → (𝑧 ∈ (𝐴𝐵) ↦ ((𝐹𝑧)𝑅(𝐺𝑧))) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3230, 31eqtrd 2655 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧𝐶 ↦ ((𝐹𝑧)𝑅(𝐺𝑧))))
3332feq1d 5992 . 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 384   = wceq 1480  wcel 1987  wral 2907  cin 3558  cmpt 4678   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑓 cof 6855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857
This theorem is referenced by: (None)
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