Theorem eldmrexrn 7092

Description: For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Assertion

Ref	Expression
eldmrexrn	⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem eldmrexrn

Step	Hyp	Ref	Expression
1		fvelrn 7078	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → (𝐹‘𝑌) ∈ ran 𝐹)
2		eqid 2731	. . 3 ⊢ (𝐹‘𝑌) = (𝐹‘𝑌)
3		eqeq1 2735	. . . 4 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = (𝐹‘𝑌)))
4	3	rspcev 3612	. . 3 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ (𝐹‘𝑌) = (𝐹‘𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))
5	1, 2, 4	sylancl 585	. 2 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))
6	5	ex 412	1 ⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  dom cdm 5676  ran crn 5677  Fun wfun 6537  ‘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-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-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-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:  eldmrexrnb  7093