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Theorem eldmrexrn 5229
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 𝐹x ran 𝐹 x = (𝐹𝑌)))
Distinct variable groups:   x,𝐹   x,𝑌

Proof of Theorem eldmrexrn
StepHypRef Expression
1 fvelrn 5219 . . 3 ((Fun 𝐹 𝑌 dom 𝐹) → (𝐹𝑌) ran 𝐹)
2 eqid 2018 . . 3 (𝐹𝑌) = (𝐹𝑌)
3 eqeq1 2024 . . . 4 (x = (𝐹𝑌) → (x = (𝐹𝑌) ↔ (𝐹𝑌) = (𝐹𝑌)))
43rspcev 2629 . . 3 (((𝐹𝑌) ran 𝐹 (𝐹𝑌) = (𝐹𝑌)) → x ran 𝐹 x = (𝐹𝑌))
51, 2, 4sylancl 394 . 2 ((Fun 𝐹 𝑌 dom 𝐹) → x ran 𝐹 x = (𝐹𝑌))
65ex 108 1 (Fun 𝐹 → (𝑌 dom 𝐹x ran 𝐹 x = (𝐹𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  wrex 2281  dom cdm 4268  ran crn 4269  Fun wfun 4819  cfv 4825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833
This theorem is referenced by: (None)
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