Theorem elrnrexdmb 5636

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 5228	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fvelrnb 5544	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌))
3	1, 2	sylbi 120	. 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌))
4		eqcom 2172	. . 3 ⊢ (𝑌 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑌)
5	4	rexbii 2477	. 2 ⊢ (∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑌)
6	3, 5	bitr4di 197	1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∃wrex 2449  dom cdm 4611  ran crn 4612  Fun wfun 5192   Fn wfn 5193  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)