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Theorem fresaun 6290
Description: The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)
Assertion
Ref Expression
fresaun ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Proof of Theorem fresaun
StepHypRef Expression
1 simp1 1159 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐹:𝐴𝐶)
2 inss1 4029 . . . 4 (𝐴𝐵) ⊆ 𝐴
3 fssres 6285 . . . 4 ((𝐹:𝐴𝐶 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
41, 2, 3sylancl 576 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
5 difss 3936 . . . . 5 (𝐴𝐵) ⊆ 𝐴
6 fssres 6285 . . . . 5 ((𝐹:𝐴𝐶 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
71, 5, 6sylancl 576 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶)
8 simp2 1160 . . . . 5 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐺:𝐵𝐶)
9 difss 3936 . . . . 5 (𝐵𝐴) ⊆ 𝐵
10 fssres 6285 . . . . 5 ((𝐺:𝐵𝐶 ∧ (𝐵𝐴) ⊆ 𝐵) → (𝐺 ↾ (𝐵𝐴)):(𝐵𝐴)⟶𝐶)
118, 9, 10sylancl 576 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐺 ↾ (𝐵𝐴)):(𝐵𝐴)⟶𝐶)
12 indifdir 4085 . . . . . 6 ((𝐴𝐵) ∩ (𝐵𝐴)) = ((𝐴 ∩ (𝐵𝐴)) ∖ (𝐵 ∩ (𝐵𝐴)))
13 disjdif 4236 . . . . . . 7 (𝐴 ∩ (𝐵𝐴)) = ∅
1413difeq1i 3923 . . . . . 6 ((𝐴 ∩ (𝐵𝐴)) ∖ (𝐵 ∩ (𝐵𝐴))) = (∅ ∖ (𝐵 ∩ (𝐵𝐴)))
15 0dif 4175 . . . . . 6 (∅ ∖ (𝐵 ∩ (𝐵𝐴))) = ∅
1612, 14, 153eqtri 2832 . . . . 5 ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅
1716a1i 11 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅)
18 fun2 6282 . . . 4 ((((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶 ∧ (𝐺 ↾ (𝐵𝐴)):(𝐵𝐴)⟶𝐶) ∧ ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))):((𝐴𝐵) ∪ (𝐵𝐴))⟶𝐶)
197, 11, 17, 18syl21anc 857 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))):((𝐴𝐵) ∪ (𝐵𝐴))⟶𝐶)
20 indi 4075 . . . . 5 ((𝐴𝐵) ∩ ((𝐴𝐵) ∪ (𝐵𝐴))) = (((𝐴𝐵) ∩ (𝐴𝐵)) ∪ ((𝐴𝐵) ∩ (𝐵𝐴)))
21 inass 4020 . . . . . . 7 ((𝐴𝐵) ∩ (𝐴𝐵)) = (𝐴 ∩ (𝐵 ∩ (𝐴𝐵)))
22 disjdif 4236 . . . . . . . 8 (𝐵 ∩ (𝐴𝐵)) = ∅
2322ineq2i 4010 . . . . . . 7 (𝐴 ∩ (𝐵 ∩ (𝐴𝐵))) = (𝐴 ∩ ∅)
24 in0 4166 . . . . . . 7 (𝐴 ∩ ∅) = ∅
2521, 23, 243eqtri 2832 . . . . . 6 ((𝐴𝐵) ∩ (𝐴𝐵)) = ∅
26 incom 4004 . . . . . . . 8 (𝐴𝐵) = (𝐵𝐴)
2726ineq1i 4009 . . . . . . 7 ((𝐴𝐵) ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ (𝐵𝐴))
28 inass 4020 . . . . . . . 8 ((𝐵𝐴) ∩ (𝐵𝐴)) = (𝐵 ∩ (𝐴 ∩ (𝐵𝐴)))
2913ineq2i 4010 . . . . . . . 8 (𝐵 ∩ (𝐴 ∩ (𝐵𝐴))) = (𝐵 ∩ ∅)
30 in0 4166 . . . . . . . 8 (𝐵 ∩ ∅) = ∅
3128, 29, 303eqtri 2832 . . . . . . 7 ((𝐵𝐴) ∩ (𝐵𝐴)) = ∅
3227, 31eqtri 2828 . . . . . 6 ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅
3325, 32uneq12i 3964 . . . . 5 (((𝐴𝐵) ∩ (𝐴𝐵)) ∪ ((𝐴𝐵) ∩ (𝐵𝐴))) = (∅ ∪ ∅)
34 un0 4165 . . . . 5 (∅ ∪ ∅) = ∅
3520, 33, 343eqtri 2832 . . . 4 ((𝐴𝐵) ∩ ((𝐴𝐵) ∪ (𝐵𝐴))) = ∅
3635a1i 11 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐴𝐵) ∩ ((𝐴𝐵) ∪ (𝐵𝐴))) = ∅)
37 fun2 6282 . . 3 ((((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)⟶𝐶 ∧ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴))):((𝐴𝐵) ∪ (𝐵𝐴))⟶𝐶) ∧ ((𝐴𝐵) ∩ ((𝐴𝐵) ∪ (𝐵𝐴))) = ∅) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))⟶𝐶)
384, 19, 36, 37syl21anc 857 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))⟶𝐶)
39 ffn 6256 . . . . 5 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
40 ffn 6256 . . . . 5 (𝐺:𝐵𝐶𝐺 Fn 𝐵)
41 id 22 . . . . 5 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
42 resasplit 6289 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
4339, 40, 41, 42syl3an 1192 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))
4443feq1d 6241 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺):(𝐴𝐵)⟶𝐶 ↔ ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):(𝐴𝐵)⟶𝐶))
45 un12 3970 . . . . 5 ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))
4626uneq1i 3962 . . . . . . 7 ((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ (𝐵𝐴))
47 inundif 4242 . . . . . . 7 ((𝐵𝐴) ∪ (𝐵𝐴)) = 𝐵
4846, 47eqtri 2828 . . . . . 6 ((𝐴𝐵) ∪ (𝐵𝐴)) = 𝐵
4948uneq2i 3963 . . . . 5 ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴))) = ((𝐴𝐵) ∪ 𝐵)
50 undif1 4239 . . . . 5 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
5145, 49, 503eqtri 2832 . . . 4 ((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴))) = (𝐴𝐵)
5251feq2i 6248 . . 3 (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))⟶𝐶 ↔ ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):(𝐴𝐵)⟶𝐶)
5344, 52syl6rbbr 281 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))):((𝐴𝐵) ∪ ((𝐴𝐵) ∪ (𝐵𝐴)))⟶𝐶 ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
5438, 53mpbid 223 1 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1100   = wceq 1637  cdif 3766  cun 3767  cin 3768  wss 3769  c0 4116  cres 5313   Fn wfn 6096  wf 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-fun 6103  df-fn 6104  df-f 6105
This theorem is referenced by:  cvmliftlem10  31599  elmapresaun  37836
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