Theorem fnunirn 5735

Description: Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
fnunirn	⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐼   𝑥,𝐹

Proof of Theorem fnunirn

Step	Hyp	Ref	Expression
1		fnfun 5285	. . 3 ⊢ (𝐹 Fn 𝐼 → Fun 𝐹)
2		elunirn 5734	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
3	1, 2	syl 14	. 2 ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
4		fndm 5287	. . 3 ⊢ (𝐹 Fn 𝐼 → dom 𝐹 = 𝐼)
5	4	rexeqdv 2668	. 2 ⊢ (𝐹 Fn 𝐼 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥)))
6	3, 5	bitrd 187	1 ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2136  ∃wrex 2445  ∪ cuni 3789  dom cdm 4604  ran crn 4605  Fun wfun 5182   Fn wfn 5183  ‘cfv 5188

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  xmetunirn  12998