Theorem elrnmpt1 4798

Description: Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
elrnmpt1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)

Proof of Theorem elrnmpt1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2692	. . . 4 ⊢ 𝑥 ∈ V
2		id 19	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3		csbeq1a 3016	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
4	2, 3	eleq12d 2211	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
5		csbeq1a 3016	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
6	5	biantrud 302	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
7	4, 6	bitr2d 188	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ↔ 𝑥 ∈ 𝐴))
8	7	equcoms 1685	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ↔ 𝑥 ∈ 𝐴))
9	1, 8	spcev 2784	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10		df-rex 2423	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
11		nfv 1509	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)
12		nfcsb1v 3040	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
13	12	nfcri 2276	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
14		nfcsb1v 3040	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfeq2 2294	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1545	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
17	5	eqeq2d 2152	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
18	4, 17	anbi12d 465	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)))
19	11, 16, 18	cbvex 1730	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
20	10, 19	bitri 183	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
21		eqeq1 2147	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
22	21	anbi2d 460	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
23	22	exbidv 1798	. . . . 5 ⊢ (𝑦 = 𝐵 → (∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
24	20, 23	syl5bb 191	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
25		rnmpt.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	rnmpt 4795	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
27	24, 26	elab2g 2835	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
28	9, 27	syl5ibr 155	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹))
29	28	impcom 124	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃wrex 2418  ⦋csb 3007   ↦ cmpt 3997  ran crn 4548

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-mpt 3999  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  fliftel1  5703