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Theorem elrnrexdm 6361
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 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem elrnrexdm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2622 . . . . . 6 (𝑌 ∈ ran 𝐹𝑌 = 𝑌)
21ancli 574 . . . . 5 (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
32adantl 482 . . . 4 ((Fun 𝐹𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
4 eqeq2 2632 . . . . 5 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
54rspcev 3307 . . . 4 ((𝑌 ∈ ran 𝐹𝑌 = 𝑌) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
63, 5syl 17 . . 3 ((Fun 𝐹𝑌 ∈ ran 𝐹) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
76ex 450 . 2 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦))
8 funfn 5916 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
9 eqeq2 2632 . . . 4 (𝑦 = (𝐹𝑥) → (𝑌 = 𝑦𝑌 = (𝐹𝑥)))
109rexrn 6359 . . 3 (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
118, 10sylbi 207 . 2 (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
127, 11sylibd 229 1 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wrex 2912  dom cdm 5112  ran crn 5113  Fun wfun 5880   Fn wfn 5881  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894
This theorem is referenced by:  toprntopon  20723  wlkiswwlksupgr2  26757  bj-ccinftydisj  33080  gneispace  38258
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