Theorem eldmrexrn 5706

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 5696	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → (𝐹‘𝑌) ∈ ran 𝐹)
2		eqid 2196	. . 3 ⊢ (𝐹‘𝑌) = (𝐹‘𝑌)
3		eqeq1 2203	. . . 4 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = (𝐹‘𝑌)))
4	3	rspcev 2868	. . 3 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ (𝐹‘𝑌) = (𝐹‘𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))
5	1, 2, 4	sylancl 413	. 2 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))
6	5	ex 115	1 ⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  dom cdm 4664  ran crn 4665  Fun wfun 5253  ‘cfv 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267

This theorem is referenced by: (None)