Theorem fvun2 7014

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 4181	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	fveq1i 6921	. 2 ⊢ ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐺 ∪ 𝐹)‘𝑋)
3		incom 4230	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
4	3	eqeq1i 2745	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
5	4	anbi1i 623	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵) ↔ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵))
6		fvun1 7013	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
7	5, 6	syl3an3b 1405	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
8	7	3com12 1123	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
9	2, 8	eqtrid 2792	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   ∩ cin 3975  ∅c0 4352   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  fvun2d  7016  fveqf1o  7338  frrlem12  8338  ptunhmeo  23837  noextenddif  27731  noextendlt  27732  noextendgt  27733  noetasuplem4  27799  axlowdimlem9  28983  axlowdimlem12  28986  axlowdimlem17  28991  vtxdun  29517  isoun  32713  resf1o  32744  cycpmfvlem  33105  elrspunidl  33421  lbsdiflsp0  33639  sseqfv2  34359  actfunsnrndisj  34582  reprsuc  34592  breprexplema  34607  cvmliftlem4  35256  fullfunfv  35911  finixpnum  37565  poimirlem1  37581  poimirlem2  37582  poimirlem3  37583  poimirlem4  37584  poimirlem6  37586  poimirlem7  37587  poimirlem11  37591  poimirlem12  37592  poimirlem16  37596  poimirlem19  37599  poimirlem20  37600  poimirlem23  37603  poimirlem28  37608