Theorem eluniimadm 5971

Description: Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)

Assertion

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

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

Proof of Theorem eluniimadm

Step	Hyp	Ref	Expression
1		funiunfvdm 5969	. . 3 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))
2	1	eleq2d 2308	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ↔ 𝐵 ∈ ∪ (𝐹 “ 𝐴)))
3		eliun 4016	. 2 ⊢ (𝐵 ∈ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥))
4	2, 3	bitr3di 195	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ∪ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∈ wcel 2209  ∃wrex 2529  ∪ cuni 3935  ∪ ciun 4012   “ cima 4777   Fn wfn 5372  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  suplocexprlemell  8080