Theorem elrnrexdm 5249

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 = (𝐹‘x)))

Distinct variable groups:   x,𝐹   x,𝑌

Proof of Theorem elrnrexdm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2038	. . . . . 6 ⊢ (𝑌 ∈ ran 𝐹 → 𝑌 = 𝑌)
2	1	ancli 306	. . . . 5 ⊢ (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌))
3	2	adantl 262	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌))
4		eqeq2 2046	. . . . 5 ⊢ (y = 𝑌 → (𝑌 = y ↔ 𝑌 = 𝑌))
5	4	rspcev 2650	. . . 4 ⊢ ((𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌) → ∃y ∈ ran 𝐹 𝑌 = y)
6	3, 5	syl 14	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → ∃y ∈ ran 𝐹 𝑌 = y)
7	6	ex 108	. 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃y ∈ ran 𝐹 𝑌 = y))
8		funfn 4874	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
9		eqeq2 2046	. . . 4 ⊢ (y = (𝐹‘x) → (𝑌 = y ↔ 𝑌 = (𝐹‘x)))
10	9	rexrn 5247	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (∃y ∈ ran 𝐹 𝑌 = y ↔ ∃x ∈ dom 𝐹 𝑌 = (𝐹‘x)))
11	8, 10	sylbi 114	. 2 ⊢ (Fun 𝐹 → (∃y ∈ ran 𝐹 𝑌 = y ↔ ∃x ∈ dom 𝐹 𝑌 = (𝐹‘x)))
12	7, 11	sylibd 138	1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃x ∈ dom 𝐹 𝑌 = (𝐹‘x)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by: (None)