Theorem eluniima 7266

Description: Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)

Assertion

Ref	Expression
eluniima	⊢ (Fun 𝐹 → (𝐵 ∈ ∪ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥)))

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

Proof of Theorem eluniima

Step	Hyp	Ref	Expression
1		funiunfv 7264	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))
2	1	eleq2d 2815	. 2 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ↔ 𝐵 ∈ ∪ (𝐹 “ 𝐴)))
3		eliun 5004	. 2 ⊢ (𝐵 ∈ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥))
4	2, 3	bitr3di 285	1 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2098  ∃wrex 3067  ∪ cuni 4912  ∪ ciun 5000   “ cima 5685  Fun wfun 6547  ‘cfv 6553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561

This theorem is referenced by:  elunirnALT  7268  ttrclse  9758  alephfp  10139  acsficl2d  18551  elhf  35803