Theorem elrnmptg 4513

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 4509	. . 3 ⊢ ran 𝐹 = {y ∣ ∃x ∈ A y = B}
3	2	eleq2i 2086	. 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {y ∣ ∃x ∈ A y = B})
4		r19.29 2428	. . . . 5 ⊢ ((∀x ∈ A B ∈ 𝑉 ∧ ∃x ∈ A 𝐶 = B) → ∃x ∈ A (B ∈ 𝑉 ∧ 𝐶 = B))
5		eleq1 2082	. . . . . . . 8 ⊢ (𝐶 = B → (𝐶 ∈ 𝑉 ↔ B ∈ 𝑉))
6	5	biimparc 283	. . . . . . 7 ⊢ ((B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ 𝑉)
7		elex 2543	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
8	6, 7	syl 14	. . . . . 6 ⊢ ((B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ V)
9	8	rexlimivw 2407	. . . . 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 2028	. . . . 5 ⊢ (y = 𝐶 → (y = B ↔ 𝐶 = B))
13	12	rexbidv 2305	. . . 4 ⊢ (y = 𝐶 → (∃x ∈ A y = B ↔ ∃x ∈ A 𝐶 = B))
14	13	elab3g 2670	. . 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 1228   ∈ wcel 1374  {cab 2008  ∀wral 2284  ∃wrex 2285  Vcvv 2535   ↦ cmpt 3792  ran crn 4273

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-mpt 3794  df-cnv 4280  df-dm 4282  df-rn 4283

This theorem is referenced by:  elrnmpti  4514  fliftel  5358