Theorem fnunirn 7256

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 6649	. . 3 ⊢ (𝐹 Fn 𝐼 → Fun 𝐹)
2		elunirn 7253	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
3	1, 2	syl 17	. 2 ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
4		fndm 6652	. . 3 ⊢ (𝐹 Fn 𝐼 → dom 𝐹 = 𝐼)
5	4	rexeqdv 3325	. 2 ⊢ (𝐹 Fn 𝐼 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥)))
6	3, 5	bitrd 279	1 ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2105  ∃wrex 3069  ∪ cuni 4908  dom cdm 5676  ran crn 5677  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  itunitc  10422  wunex2  10739  mreunirn  17552  arwhoma  18005  filunirn  23707  xmetunirn  24164  abfmpunirn  32312  cmpcref  33296  neibastop2lem  35712  stoweidlem59  45237