Theorem fvun2 6975

Description: The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Assertion

Ref	Expression
fvun2	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Proof of Theorem fvun2

Step	Hyp	Ref	Expression
1		uncom 4113	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	fveq1i 6884	. 2 ⊢ ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐺 ∪ 𝐹)‘𝑋)
3		incom 4163	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
4	3	eqeq1i 2768	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
5	4	anbi1i 635	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵) ↔ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵))
6		fvun1 6974	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
7	5, 6	syl3an3b 1432	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
8	7	3com12 1141	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
9	2, 8	eqtrid 2810	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∪ cun 3904   ∩ cin 3905  ∅c0 4287   Fn wfn 6533  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is referenced by:  fvun2d  6977  fveqf1o  7302  frrlem12  8295  ptunhmeo  23946  noextenddif  27810  noextendlt  27811  noextendgt  27812  noetasuplem4  27878  axlowdimlem9  29278  axlowdimlem12  29281  axlowdimlem17  29286  vtxdun  29809  isoun  33025  resf1o  33053  cycpmfvlem  33410  elrspunidl  33714  lbsdiflsp0  33994  sseqfv2  34762  actfunsnrndisj  34970  reprsuc  34980  breprexplema  34995  cvmliftlem4  35758  fullfunfv  36417  finixpnum  38234  poimirlem1  38250  poimirlem2  38251  poimirlem3  38252  poimirlem4  38253  poimirlem6  38255  poimirlem7  38256  poimirlem11  38260  poimirlem12  38261  poimirlem16  38265  poimirlem19  38268  poimirlem20  38269  poimirlem23  38272  poimirlem28  38277