Theorem elunirn2 7126

Description: Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)

Assertion

Ref	Expression
elunirn2	⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹)

Proof of Theorem elunirn2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6806	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐴 ∈ dom 𝐹)
2		fveq2 6774	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	eleq2d 2824	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ (𝐹‘𝐴)))
4	3	rspcev 3561	. . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
5	1, 4	mpancom 685	. . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
6	5	adantl 482	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
7		elunirn 7124	. . 3 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)))
8	7	adantr 481	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)))
9	6, 8	mpbird 256	1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  ∪ cuni 4839  dom cdm 5589  ran crn 5590  Fun wfun 6427  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  measbasedom  32170  sxbrsigalem0  32238  prjcrv0  40470