Theorem elrnrexdm 5552

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

Assertion

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

Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem elrnrexdm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2138	. . . . . 6 ⊢ (𝑌 ∈ ran 𝐹 → 𝑌 = 𝑌)
2	1	ancli 321	. . . . 5 ⊢ (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌))
3	2	adantl 275	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌))
4		eqeq2 2147	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑌 = 𝑦 ↔ 𝑌 = 𝑌))
5	4	rspcev 2784	. . . 4 ⊢ ((𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
6	3, 5	syl 14	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
7	6	ex 114	. 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦))
8		funfn 5148	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
9		eqeq2 2147	. . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑌 = 𝑦 ↔ 𝑌 = (𝐹‘𝑥)))
10	9	rexrn 5550	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))
11	8, 10	sylbi 120	. 2 ⊢ (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))
12	7, 11	sylibd 148	1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∃wrex 2415  dom cdm 4534  ran crn 4535  Fun wfun 5112   Fn wfn 5113  ‘cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  ennnfonelemrnh  11918  ennnfonelemf1  11920  exmidsbthrlem  13206