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Theorem elrnrexdm 6612
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 2826 . . . . . 6 (𝑌 ∈ ran 𝐹𝑌 = 𝑌)
21ancli 546 . . . . 5 (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
32adantl 475 . . . 4 ((Fun 𝐹𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
4 eqeq2 2836 . . . . 5 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
54rspcev 3526 . . . 4 ((𝑌 ∈ ran 𝐹𝑌 = 𝑌) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
63, 5syl 17 . . 3 ((Fun 𝐹𝑌 ∈ ran 𝐹) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
76ex 403 . 2 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦))
8 funfn 6153 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
9 eqeq2 2836 . . . 4 (𝑦 = (𝐹𝑥) → (𝑌 = 𝑦𝑌 = (𝐹𝑥)))
109rexrn 6610 . . 3 (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
118, 10sylbi 209 . 2 (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
127, 11sylibd 231 1 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wrex 3118  dom cdm 5342  ran crn 5343  Fun wfun 6117   Fn wfn 6118  cfv 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131
This theorem is referenced by:  toprntopon  21100  wlkiswwlksupgr2  27176  bj-ccinftydisj  33640  gneispace  39272
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