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Theorem ofun 39403
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 6439 . . 3 (𝜑 → (𝐴𝐶) Fn (𝑀𝑁))
5 ofun.b . . . 4 (𝜑𝐵 Fn 𝑀)
6 ofun.d . . . 4 (𝜑𝐷 Fn 𝑁)
75, 6, 3fnund 6439 . . 3 (𝜑 → (𝐵𝐷) Fn (𝑀𝑁))
8 ofun.m . . . 4 (𝜑𝑀𝑉)
9 ofun.n . . . 4 (𝜑𝑁𝑊)
108, 9unexd 39392 . . 3 (𝜑 → (𝑀𝑁) ∈ V)
11 inidm 4148 . . 3 ((𝑀𝑁) ∩ (𝑀𝑁)) = (𝑀𝑁)
124, 7, 10, 10, 11offn 7404 . 2 (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) Fn (𝑀𝑁))
13 inidm 4148 . . . 4 (𝑀𝑀) = 𝑀
141, 5, 8, 8, 13offn 7404 . . 3 (𝜑 → (𝐴f 𝑅𝐵) Fn 𝑀)
15 inidm 4148 . . . 4 (𝑁𝑁) = 𝑁
162, 6, 9, 9, 15offn 7404 . . 3 (𝜑 → (𝐶f 𝑅𝐷) Fn 𝑁)
1714, 16, 3fnund 6439 . 2 (𝜑 → ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)) Fn (𝑀𝑁))
18 eqidd 2802 . . . 4 ((𝜑𝑥 ∈ (𝑀𝑁)) → ((𝐴𝐶)‘𝑥) = ((𝐴𝐶)‘𝑥))
19 eqidd 2802 . . . 4 ((𝜑𝑥 ∈ (𝑀𝑁)) → ((𝐵𝐷)‘𝑥) = ((𝐵𝐷)‘𝑥))
204, 7, 10, 10, 11, 18, 19ofval 7402 . . 3 ((𝜑𝑥 ∈ (𝑀𝑁)) → (((𝐴𝐶) ∘f 𝑅(𝐵𝐷))‘𝑥) = (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)))
21 elun 4079 . . . 4 (𝑥 ∈ (𝑀𝑁) ↔ (𝑥𝑀𝑥𝑁))
22 eqidd 2802 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐴𝑥) = (𝐴𝑥))
23 eqidd 2802 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐵𝑥) = (𝐵𝑥))
241, 5, 8, 8, 13, 22, 23ofval 7402 . . . . . 6 ((𝜑𝑥𝑀) → ((𝐴f 𝑅𝐵)‘𝑥) = ((𝐴𝑥)𝑅(𝐵𝑥)))
2514adantr 484 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐴f 𝑅𝐵) Fn 𝑀)
2616adantr 484 . . . . . . 7 ((𝜑𝑥𝑀) → (𝐶f 𝑅𝐷) Fn 𝑁)
273adantr 484 . . . . . . 7 ((𝜑𝑥𝑀) → (𝑀𝑁) = ∅)
28 simpr 488 . . . . . . 7 ((𝜑𝑥𝑀) → 𝑥𝑀)
2925, 26, 27, 28fvun1d 6735 . . . . . 6 ((𝜑𝑥𝑀) → (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥) = ((𝐴f 𝑅𝐵)‘𝑥))
301adantr 484 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐴 Fn 𝑀)
312adantr 484 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐶 Fn 𝑁)
3230, 31, 27, 28fvun1d 6735 . . . . . . 7 ((𝜑𝑥𝑀) → ((𝐴𝐶)‘𝑥) = (𝐴𝑥))
335adantr 484 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐵 Fn 𝑀)
346adantr 484 . . . . . . . 8 ((𝜑𝑥𝑀) → 𝐷 Fn 𝑁)
3533, 34, 27, 28fvun1d 6735 . . . . . . 7 ((𝜑𝑥𝑀) → ((𝐵𝐷)‘𝑥) = (𝐵𝑥))
3632, 35oveq12d 7157 . . . . . 6 ((𝜑𝑥𝑀) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = ((𝐴𝑥)𝑅(𝐵𝑥)))
3724, 29, 363eqtr4rd 2847 . . . . 5 ((𝜑𝑥𝑀) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
38 eqidd 2802 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐶𝑥) = (𝐶𝑥))
39 eqidd 2802 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐷𝑥) = (𝐷𝑥))
402, 6, 9, 9, 15, 38, 39ofval 7402 . . . . . 6 ((𝜑𝑥𝑁) → ((𝐶f 𝑅𝐷)‘𝑥) = ((𝐶𝑥)𝑅(𝐷𝑥)))
4114adantr 484 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐴f 𝑅𝐵) Fn 𝑀)
4216adantr 484 . . . . . . 7 ((𝜑𝑥𝑁) → (𝐶f 𝑅𝐷) Fn 𝑁)
433adantr 484 . . . . . . 7 ((𝜑𝑥𝑁) → (𝑀𝑁) = ∅)
44 simpr 488 . . . . . . 7 ((𝜑𝑥𝑁) → 𝑥𝑁)
4541, 42, 43, 44fvun2d 6736 . . . . . 6 ((𝜑𝑥𝑁) → (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥) = ((𝐶f 𝑅𝐷)‘𝑥))
461adantr 484 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐴 Fn 𝑀)
472adantr 484 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐶 Fn 𝑁)
4846, 47, 43, 44fvun2d 6736 . . . . . . 7 ((𝜑𝑥𝑁) → ((𝐴𝐶)‘𝑥) = (𝐶𝑥))
495adantr 484 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐵 Fn 𝑀)
506adantr 484 . . . . . . . 8 ((𝜑𝑥𝑁) → 𝐷 Fn 𝑁)
5149, 50, 43, 44fvun2d 6736 . . . . . . 7 ((𝜑𝑥𝑁) → ((𝐵𝐷)‘𝑥) = (𝐷𝑥))
5248, 51oveq12d 7157 . . . . . 6 ((𝜑𝑥𝑁) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = ((𝐶𝑥)𝑅(𝐷𝑥)))
5340, 45, 523eqtr4rd 2847 . . . . 5 ((𝜑𝑥𝑁) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5437, 53jaodan 955 . . . 4 ((𝜑 ∧ (𝑥𝑀𝑥𝑁)) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5521, 54sylan2b 596 . . 3 ((𝜑𝑥 ∈ (𝑀𝑁)) → (((𝐴𝐶)‘𝑥)𝑅((𝐵𝐷)‘𝑥)) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5620, 55eqtrd 2836 . 2 ((𝜑𝑥 ∈ (𝑀𝑁)) → (((𝐴𝐶) ∘f 𝑅(𝐵𝐷))‘𝑥) = (((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷))‘𝑥))
5712, 17, 56eqfnfvd 6786 1 (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) = ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844   = wceq 1538  wcel 2112  Vcvv 3444  cun 3882  cin 3883  c0 4246   Fn wfn 6323  cfv 6328  (class class class)co 7139  f cof 7391
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393
This theorem is referenced by:  fsuppssind  39446
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