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Theorem resasplitss 5012
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 A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → ((𝐹 ↾ (AB)) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) ⊆ (𝐹𝐺))

Proof of Theorem resasplitss
StepHypRef Expression
1 unidm 3080 . . . 4 ((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) = (𝐹 ↾ (AB))
21uneq1i 3087 . . 3 (((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) = ((𝐹 ↾ (AB)) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA))))
3 un4 3097 . . . 4 (((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) = (((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA))))
4 simp3 905 . . . . . . 7 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → (𝐹 ↾ (AB)) = (𝐺 ↾ (AB)))
54uneq1d 3090 . . . . . 6 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA))) = ((𝐺 ↾ (AB)) ∪ (𝐺 ↾ (BA))))
65uneq2d 3091 . . . . 5 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → (((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) = (((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ∪ ((𝐺 ↾ (AB)) ∪ (𝐺 ↾ (BA)))))
7 resundi 4568 . . . . . . 7 (𝐹 ↾ ((AB) ∪ (AB))) = ((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB)))
8 inundifss 3295 . . . . . . . 8 ((AB) ∪ (AB)) ⊆ A
9 ssres2 4581 . . . . . . . 8 (((AB) ∪ (AB)) ⊆ A → (𝐹 ↾ ((AB) ∪ (AB))) ⊆ (𝐹A))
108, 9ax-mp 7 . . . . . . 7 (𝐹 ↾ ((AB) ∪ (AB))) ⊆ (𝐹A)
117, 10eqsstr3i 2970 . . . . . 6 ((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ⊆ (𝐹A)
12 resundi 4568 . . . . . . 7 (𝐺 ↾ ((AB) ∪ (BA))) = ((𝐺 ↾ (AB)) ∪ (𝐺 ↾ (BA)))
13 incom 3123 . . . . . . . . . 10 (AB) = (BA)
1413uneq1i 3087 . . . . . . . . 9 ((AB) ∪ (BA)) = ((BA) ∪ (BA))
15 inundifss 3295 . . . . . . . . 9 ((BA) ∪ (BA)) ⊆ B
1614, 15eqsstri 2969 . . . . . . . 8 ((AB) ∪ (BA)) ⊆ B
17 ssres2 4581 . . . . . . . 8 (((AB) ∪ (BA)) ⊆ B → (𝐺 ↾ ((AB) ∪ (BA))) ⊆ (𝐺B))
1816, 17ax-mp 7 . . . . . . 7 (𝐺 ↾ ((AB) ∪ (BA))) ⊆ (𝐺B)
1912, 18eqsstr3i 2970 . . . . . 6 ((𝐺 ↾ (AB)) ∪ (𝐺 ↾ (BA))) ⊆ (𝐺B)
20 unss12 3109 . . . . . 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 2989 . . . 4 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → (((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) ⊆ ((𝐹A) ∪ (𝐺B)))
233, 22syl5eqssr 2984 . . 3 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → (((𝐹 ↾ (AB)) ∪ (𝐹 ↾ (AB))) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) ⊆ ((𝐹A) ∪ (𝐺B)))
242, 23syl5eqssr 2984 . 2 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → ((𝐹 ↾ (AB)) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) ⊆ ((𝐹A) ∪ (𝐺B)))
25 fnresdm 4951 . . . 4 (𝐹 Fn A → (𝐹A) = 𝐹)
26 fnresdm 4951 . . . 4 (𝐺 Fn B → (𝐺B) = 𝐺)
27 uneq12 3086 . . . 4 (((𝐹A) = 𝐹 (𝐺B) = 𝐺) → ((𝐹A) ∪ (𝐺B)) = (𝐹𝐺))
2825, 26, 27syl2an 273 . . 3 ((𝐹 Fn A 𝐺 Fn B) → ((𝐹A) ∪ (𝐺B)) = (𝐹𝐺))
29283adant3 923 . 2 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → ((𝐹A) ∪ (𝐺B)) = (𝐹𝐺))
3024, 29sseqtrd 2975 1 ((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → ((𝐹 ↾ (AB)) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) ⊆ (𝐹𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   w3a 884   = wceq 1242  cdif 2908  cun 2909  cin 2910  wss 2911  cres 4290   Fn wfn 4840
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-res 4300  df-fun 4847  df-fn 4848
This theorem is referenced by: (None)
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