Theorem fvun2 5746

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 3365	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	fveq1i 5673	. 2 ⊢ ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐺 ∪ 𝐹)‘𝑋)
3		incom 3413	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
4	3	eqeq1i 2242	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
5	4	anbi1i 458	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵) ↔ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵))
6		fvun1 5745	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐵 ∩ 𝐴) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
7	5, 6	syl3an3b 1312	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
8	7	3com12 1234	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐺 ∪ 𝐹)‘𝑋) = (𝐺‘𝑋))
9	2, 8	eqtrid 2279	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   ∪ cun 3211   ∩ cin 3212  ∅c0 3510   Fn wfn 5349  ‘cfv 5354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by:  caseinr  7385  vtxdfifiun  16341  gfsump1  16917