Theorem elrnrexdmb 5774

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 5347	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fvelrnb 5680	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌))
3	1, 2	sylbi 121	. 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌))
4		eqcom 2231	. . 3 ⊢ (𝑌 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑌)
5	4	rexbii 2537	. 2 ⊢ (∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌)
6	3, 5	bitr4di 198	1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  dom cdm 4718  ran crn 4719  Fun wfun 5311   Fn wfn 5312  ‘cfv 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  edgiedgbg  15859