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Theorem fun 5263
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fun (((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))

Proof of Theorem fun
StepHypRef Expression
1 fnun 5197 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
21expcom 115 . . . 4 ((𝐴𝐵) = ∅ → ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹𝐺) Fn (𝐴𝐵)))
3 rnun 4915 . . . . . 6 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
4 unss12 3216 . . . . . 6 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶𝐷))
53, 4eqsstrid 3111 . . . . 5 ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷))
65a1i 9 . . . 4 ((𝐴𝐵) = ∅ → ((ran 𝐹𝐶 ∧ ran 𝐺𝐷) → ran (𝐹𝐺) ⊆ (𝐶𝐷)))
72, 6anim12d 331 . . 3 ((𝐴𝐵) = ∅ → (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)) → ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷))))
8 df-f 5095 . . . . 5 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
9 df-f 5095 . . . . 5 (𝐺:𝐵𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷))
108, 9anbi12i 453 . . . 4 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)))
11 an4 558 . . . 4 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐷)) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
1210, 11bitri 183 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐷) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (ran 𝐹𝐶 ∧ ran 𝐺𝐷)))
13 df-f 5095 . . 3 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐵) ∧ ran (𝐹𝐺) ⊆ (𝐶𝐷)))
147, 12, 133imtr4g 204 . 2 ((𝐴𝐵) = ∅ → ((𝐹:𝐴𝐶𝐺:𝐵𝐷) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷)))
1514impcom 124 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  cun 3037  cin 3038  wss 3039  c0 3331  ran crn 4508   Fn wfn 5086  wf 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-f 5095
This theorem is referenced by:  fun2  5264  ftpg  5570  fsnunf  5586
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