Theorem eldmrexrn 7038

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 7023	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → (𝐹‘𝑌) ∈ ran 𝐹)
2		eqid 2737	. . 3 ⊢ (𝐹‘𝑌) = (𝐹‘𝑌)
3		eqeq1 2741	. . . 4 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = (𝐹‘𝑌)))
4	3	rspcev 3565	. . 3 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ (𝐹‘𝑌) = (𝐹‘𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))
5	1, 2, 4	sylancl 587	. 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 1542   ∈ wcel 2114  ∃wrex 3062  dom cdm 5625  ran crn 5626  Fun wfun 6487  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501

This theorem is referenced by:  eldmrexrnb  7039