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Theorem resasplitss 5396
Description: If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
Assertion
Ref Expression
resasplitss ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ (𝐹𝐺))

Proof of Theorem resasplitss
StepHypRef Expression
1 unidm 3279 . . . 4 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
21uneq1i 3286 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
3 un4 3296 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
4 simp3 999 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
54uneq1d 3289 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
65uneq2d 3290 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
7 resundi 4921 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵)))
8 inundifss 3501 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
9 ssres2 4935 . . . . . . . 8 (((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴 → (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) ⊆ (𝐹𝐴))
108, 9ax-mp 5 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) ⊆ (𝐹𝐴)
117, 10eqsstrri 3189 . . . . . 6 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ⊆ (𝐹𝐴)
12 resundi 4921 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
13 incom 3328 . . . . . . . . . 10 (𝐴𝐵) = (𝐵𝐴)
1413uneq1i 3286 . . . . . . . . 9 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
15 inundifss 3501 . . . . . . . . 9 ((𝐵𝐴) ∪ (𝐵𝐴)) ⊆ 𝐵
1614, 15eqsstri 3188 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ 𝐵
17 ssres2 4935 . . . . . . . 8 (((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ 𝐵 → (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) ⊆ (𝐺𝐵))
1816, 17ax-mp 5 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) ⊆ (𝐺𝐵)
1912, 18eqsstrri 3189 . . . . . 6 ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ⊆ (𝐺𝐵)
20 unss12 3308 . . . . . 6 ((((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ⊆ (𝐹𝐴) ∧ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ⊆ (𝐺𝐵)) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
2111, 19, 20mp2an 426 . . . . 5 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵))
226, 21eqsstrdi 3208 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
233, 22eqsstrrid 3203 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
242, 23eqsstrrid 3203 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
25 fnresdm 5326 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
26 fnresdm 5326 . . . 4 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
27 uneq12 3285 . . . 4 (((𝐹𝐴) = 𝐹 ∧ (𝐺𝐵) = 𝐺) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
2825, 26, 27syl2an 289 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
29283adant3 1017 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
3024, 29sseqtrd 3194 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ (𝐹𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 978   = wceq 1353  cdif 3127  cun 3128  cin 3129  wss 3130  cres 4629   Fn wfn 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-dm 4637  df-res 4639  df-fun 5219  df-fn 5220
This theorem is referenced by: (None)
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