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Theorem resasplitss 5069
 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 3086 . . . 4 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
21uneq1i 3093 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
3 un4 3103 . . . 4 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) = (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
4 simp3 906 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
54uneq1d 3096 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))))
65uneq2d 3097 . . . . 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 4625 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) = ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵)))
8 inundifss 3301 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
9 ssres2 4638 . . . . . . . 8 (((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴 → (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) ⊆ (𝐹𝐴))
108, 9ax-mp 7 . . . . . . 7 (𝐹 ↾ ((𝐴𝐵) ∪ (𝐴𝐵))) ⊆ (𝐹𝐴)
117, 10eqsstr3i 2976 . . . . . 6 ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ⊆ (𝐹𝐴)
12 resundi 4625 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))
13 incom 3129 . . . . . . . . . 10 (𝐴𝐵) = (𝐵𝐴)
1413uneq1i 3093 . . . . . . . . 9 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
15 inundifss 3301 . . . . . . . . 9 ((𝐵𝐴) ∪ (𝐵𝐴)) ⊆ 𝐵
1614, 15eqsstri 2975 . . . . . . . 8 ((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ 𝐵
17 ssres2 4638 . . . . . . . 8 (((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ 𝐵 → (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) ⊆ (𝐺𝐵))
1816, 17ax-mp 7 . . . . . . 7 (𝐺 ↾ ((𝐴𝐵) ∪ (𝐵𝐴))) ⊆ (𝐺𝐵)
1912, 18eqsstr3i 2976 . . . . . 6 ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ⊆ (𝐺𝐵)
20 unss12 3115 . . . . . 6 ((((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ⊆ (𝐹𝐴) ∧ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))) ⊆ (𝐺𝐵)) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
2111, 19, 20mp2an 402 . . . . 5 (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐺 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵))
226, 21syl6eqss 2995 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
233, 22syl5eqssr 2990 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ (𝐹 ↾ (𝐴𝐵))) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
242, 23syl5eqssr 2990 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ ((𝐹𝐴) ∪ (𝐺𝐵)))
25 fnresdm 5008 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
26 fnresdm 5008 . . . 4 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
27 uneq12 3092 . . . 4 (((𝐹𝐴) = 𝐹 ∧ (𝐺𝐵) = 𝐺) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
2825, 26, 27syl2an 273 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
29283adant3 924 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐴) ∪ (𝐺𝐵)) = (𝐹𝐺))
3024, 29sseqtrd 2981 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 885   = wceq 1243   ∖ cdif 2914   ∪ cun 2915   ∩ cin 2916   ⊆ wss 2917   ↾ cres 4347   Fn wfn 4897 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355  df-res 4357  df-fun 4904  df-fn 4905 This theorem is referenced by: (None)
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