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Theorem ofun 40659
Description: A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.)
Hypotheses
Ref Expression
ofun.a (𝜑𝐴 Fn 𝑀)
ofun.b (𝜑𝐵 Fn 𝑀)
ofun.c (𝜑𝐶 Fn 𝑁)
ofun.d (𝜑𝐷 Fn 𝑁)
ofun.m (𝜑𝑀𝑉)
ofun.n (𝜑𝑁𝑊)
ofun.1 (𝜑 → (𝑀𝑁) = ∅)
Assertion
Ref Expression
ofun (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) = ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)))

Proof of Theorem ofun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofun.a . . . 4 (𝜑𝐴 Fn 𝑀)
2 ofun.c . . . 4 (𝜑𝐶 Fn 𝑁)
3 ofun.1 . . . 4 (𝜑 → (𝑀𝑁) = ∅)
41, 2, 3fnund 6615 . . 3 (𝜑 → (𝐴𝐶) Fn (𝑀𝑁))
5 ofun.b . . . 4 (𝜑𝐵 Fn 𝑀)
6 ofun.d . . . 4 (𝜑𝐷 Fn 𝑁)
75, 6, 3fnund 6615 . . 3 (𝜑 → (𝐵𝐷) Fn (𝑀𝑁))
8 ofun.m . . . 4 (𝜑𝑀𝑉)
9 ofun.n . . . 4 (𝜑𝑁𝑊)
108, 9unexd 7688 . . 3 (𝜑 → (𝑀𝑁) ∈ V)
11 inidm 4178 . . 3 ((𝑀𝑁) ∩ (𝑀𝑁)) = (𝑀𝑁)
124, 7, 10, 10, 11offn 7630 . 2 (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) Fn (𝑀𝑁))
13 inidm 4178 . . . 4 (𝑀𝑀) = 𝑀
141, 5, 8, 8, 13offn 7630 . . 3 (𝜑 → (𝐴f 𝑅𝐵) Fn 𝑀)
15 inidm 4178 . . . 4 (𝑁𝑁) = 𝑁
162, 6, 9, 9, 15offn 7630 . . 3 (𝜑 → (𝐶f 𝑅𝐷) Fn 𝑁)
1714, 16, 3fnund 6615 . 2 (𝜑 → ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)) Fn (𝑀𝑁))
18 eqidd 2737 . . . 4 ((𝜑𝑥 ∈ (𝑀𝑁)) → ((𝐴𝐶)‘𝑥) = ((𝐴𝐶)‘𝑥))
19 eqidd 2737 . . . 4 ((𝜑𝑥 ∈ (𝑀𝑁)) → ((𝐵𝐷)‘𝑥) = ((𝐵𝐷)‘𝑥))
204, 7, 10, 10, 11, 18, 19ofval 7628 . . 3 ((𝜑𝑥 ∈ (𝑀𝑁)) → (((𝐴𝐶) ∘f 𝑅(𝐵𝐷))‘𝑥) = (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)))
21 elun 4108 . . . 4 (𝑥 ∈ (𝑀𝑁) ↔ (𝑥𝑀𝑥𝑁))
22 eqidd 2737 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐴𝑥) = (𝐴𝑥))
23 eqidd 2737 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐵𝑥) = (𝐵𝑥))
241, 5, 8, 8, 13, 22, 23ofval 7628 . . . . . 6 ((𝜑𝑥𝑀) → ((𝐴f 𝑅𝐵)‘𝑥) = ((𝐴𝑥)𝑅(𝐵𝑥)))
2514adantr 481 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐴f 𝑅𝐵) Fn 𝑀)
2616adantr 481 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐶f 𝑅𝐷) Fn 𝑁)
273adantr 481 . . . . . . 7 ((𝜑𝑥𝑀) → (𝑀𝑁) = ∅)
28 simpr 485 . . . . . . 7 ((𝜑𝑥𝑀) → 𝑥𝑀)
2925, 26, 27, 28fvun1d 6934 . . . . . 6 ((𝜑𝑥𝑀) → (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥) = ((𝐴f 𝑅𝐵)‘𝑥))
301adantr 481 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐴 Fn 𝑀)
312adantr 481 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐶 Fn 𝑁)
3230, 31, 27, 28fvun1d 6934 . . . . . . 7 ((𝜑𝑥𝑀) → ((𝐴𝐶)‘𝑥) = (𝐴𝑥))
335adantr 481 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐵 Fn 𝑀)
346adantr 481 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐷 Fn 𝑁)
3533, 34, 27, 28fvun1d 6934 . . . . . . 7 ((𝜑𝑥𝑀) → ((𝐵𝐷)‘𝑥) = (𝐵𝑥))
3632, 35oveq12d 7375 . . . . . 6 ((𝜑𝑥𝑀) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = ((𝐴𝑥)𝑅(𝐵𝑥)))
3724, 29, 363eqtr4rd 2787 . . . . 5 ((𝜑𝑥𝑀) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
38 eqidd 2737 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐶𝑥) = (𝐶𝑥))
39 eqidd 2737 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐷𝑥) = (𝐷𝑥))
402, 6, 9, 9, 15, 38, 39ofval 7628 . . . . . 6 ((𝜑𝑥𝑁) → ((𝐶f 𝑅𝐷)‘𝑥) = ((𝐶𝑥)𝑅(𝐷𝑥)))
4114adantr 481 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐴f 𝑅𝐵) Fn 𝑀)
4216adantr 481 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐶f 𝑅𝐷) Fn 𝑁)
433adantr 481 . . . . . . 7 ((𝜑𝑥𝑁) → (𝑀𝑁) = ∅)
44 simpr 485 . . . . . . 7 ((𝜑𝑥𝑁) → 𝑥𝑁)
4541, 42, 43, 44fvun2d 6935 . . . . . 6 ((𝜑𝑥𝑁) → (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥) = ((𝐶f 𝑅𝐷)‘𝑥))
461adantr 481 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐴 Fn 𝑀)
472adantr 481 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐶 Fn 𝑁)
4846, 47, 43, 44fvun2d 6935 . . . . . . 7 ((𝜑𝑥𝑁) → ((𝐴𝐶)‘𝑥) = (𝐶𝑥))
495adantr 481 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐵 Fn 𝑀)
506adantr 481 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐷 Fn 𝑁)
5149, 50, 43, 44fvun2d 6935 . . . . . . 7 ((𝜑𝑥𝑁) → ((𝐵𝐷)‘𝑥) = (𝐷𝑥))
5248, 51oveq12d 7375 . . . . . 6 ((𝜑𝑥𝑁) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = ((𝐶𝑥)𝑅(𝐷𝑥)))
5340, 45, 523eqtr4rd 2787 . . . . 5 ((𝜑𝑥𝑁) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5437, 53jaodan 956 . . . 4 ((𝜑 ∧ (𝑥𝑀𝑥𝑁)) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5521, 54sylan2b 594 . . 3 ((𝜑𝑥 ∈ (𝑀𝑁)) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5620, 55eqtrd 2776 . 2 ((𝜑𝑥 ∈ (𝑀𝑁)) → (((𝐴𝐶) ∘f 𝑅(𝐵𝐷))‘𝑥) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5712, 17, 56eqfnfvd 6985 1 (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) = ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  cin 3909  c0 4282   Fn wfn 6491  cfv 6496  (class class class)co 7357  f cof 7615
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617
This theorem is referenced by:  fsuppssind  40754
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