Theorem elrnmptg 4529

Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rnmpt.1	⊢ 𝐹 = (x ∈ A ↦ B)

Assertion

Ref	Expression
elrnmptg	⊢ (∀x ∈ A B ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃x ∈ A 𝐶 = B))

Distinct variable group:   x,𝐶

Allowed substitution hints:   A(x)   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem elrnmptg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnmpt.1	. . . 4 ⊢ 𝐹 = (x ∈ A ↦ B)
2	1	rnmpt 4525	. . 3 ⊢ ran 𝐹 = {y ∣ ∃x ∈ A y = B}
3	2	eleq2i 2101	. 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {y ∣ ∃x ∈ A y = B})
4		r19.29 2444	. . . . 5 ⊢ ((∀x ∈ A B ∈ 𝑉 ∧ ∃x ∈ A 𝐶 = B) → ∃x ∈ A (B ∈ 𝑉 ∧ 𝐶 = B))
5		eleq1 2097	. . . . . . . 8 ⊢ (𝐶 = B → (𝐶 ∈ 𝑉 ↔ B ∈ 𝑉))
6	5	biimparc 283	. . . . . . 7 ⊢ ((B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ 𝑉)
7		elex 2560	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
8	6, 7	syl 14	. . . . . 6 ⊢ ((B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ V)
9	8	rexlimivw 2423	. . . . 5 ⊢ (∃x ∈ A (B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ V)
10	4, 9	syl 14	. . . 4 ⊢ ((∀x ∈ A B ∈ 𝑉 ∧ ∃x ∈ A 𝐶 = B) → 𝐶 ∈ V)
11	10	ex 108	. . 3 ⊢ (∀x ∈ A B ∈ 𝑉 → (∃x ∈ A 𝐶 = B → 𝐶 ∈ V))
12		eqeq1 2043	. . . . 5 ⊢ (y = 𝐶 → (y = B ↔ 𝐶 = B))
13	12	rexbidv 2321	. . . 4 ⊢ (y = 𝐶 → (∃x ∈ A y = B ↔ ∃x ∈ A 𝐶 = B))
14	13	elab3g 2687	. . 3 ⊢ ((∃x ∈ A 𝐶 = B → 𝐶 ∈ V) → (𝐶 ∈ {y ∣ ∃x ∈ A y = B} ↔ ∃x ∈ A 𝐶 = B))
15	11, 14	syl 14	. 2 ⊢ (∀x ∈ A B ∈ 𝑉 → (𝐶 ∈ {y ∣ ∃x ∈ A y = B} ↔ ∃x ∈ A 𝐶 = B))
16	3, 15	syl5bb 181	1 ⊢ (∀x ∈ A B ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃x ∈ A 𝐶 = B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ↦ cmpt 3809  ran crn 4289

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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by:  elrnmpti  4530  fliftel  5376