Theorem elrnmpt1 5918

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 3450	. . . 4 ⊢ 𝑥 ∈ V
2		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3		csbeq1a 3872	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
4	2, 3	eleq12d 2826	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
5		csbeq1a 3872	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
6	5	biantrud 532	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
7	4, 6	bitr2d 279	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ↔ 𝑥 ∈ 𝐴))
8	7	equcoms 2023	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ↔ 𝑥 ∈ 𝐴))
9	1, 8	spcev 3566	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10		df-rex 3070	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
11		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)
12		nfcsb1v 3883	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
13	12	nfcri 2889	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
14		nfcsb1v 3883	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfeq2 2919	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
17	5	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
18	4, 17	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)))
19	11, 16, 18	cbvexv1 2338	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
20	10, 19	bitri 274	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
21		eqeq1 2735	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
22	21	anbi2d 629	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
23	22	exbidv 1924	. . . . 5 ⊢ (𝑦 = 𝐵 → (∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
24	20, 23	bitrid 282	. . . 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 5915	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
27	24, 26	elab2g 3635	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
28	9, 27	imbitrrid 245	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹))
29	28	impcom 408	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3069  ⦋csb 3858   ↦ cmpt 5193  ran crn 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-mpt 5194  df-cnv 5646  df-dm 5648  df-rn 5649

This theorem is referenced by:  fliftel1  7260  minveclem4  24833  minvecolem4  29885  rexunirn  31484  esum2d  32781  exrecfnlem  35923  totbndbnd  36321  rrnequiv  36367  suprnmpt  43513  disjf1o  43532  disjinfi  43534  choicefi  43542  elrnmpt1d  43576  suprubrnmpt  43602  supxrleubrnmpt  43761  suprleubrnmpt  43777  infrnmptle  43778  infxrunb3rnmpt  43783  supminfrnmpt  43800  infxrgelbrnmpt  43809  fourierdlem31  44499  ioorrnopnlem  44665  sge0f1o  44743  sge0supre  44750  sge0gerp  44756  sge0iunmpt  44779  sge0rernmpt  44783  sge0reuz  44808  meadjiunlem  44826  iunhoiioolem  45036  vonioolem1  45041  smfpimcclem  45168