Theorem elrnrexdmb 7109

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, 17-Dec-2017.)

Assertion

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

Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem elrnrexdmb

Step	Hyp	Ref	Expression
1		funfn 6595	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fvelrnb 6968	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌))
3	1, 2	sylbi 217	. 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌))
4		eqcom 2743	. . 3 ⊢ (𝑌 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑌)
5	4	rexbii 3093	. 2 ⊢ (∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌)
6	3, 5	bitr4di 289	1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  ∃wrex 3069  dom cdm 5684  ran crn 5685  Fun wfun 6554   Fn wfn 6555  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568

This theorem is referenced by:  edgiedgb  29072  uhgrspansubgrlem  29308  cycpmrn  33164