Theorem elrnmpt1 5856

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 3426	. . . 4 ⊢ 𝑥 ∈ V
2		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3		csbeq1a 3842	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
4	2, 3	eleq12d 2833	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
5		csbeq1a 3842	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
6	5	biantrud 531	. . . . . 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 2024	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ↔ 𝑥 ∈ 𝐴))
9	1, 8	spcev 3535	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10		df-rex 3069	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
11		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)
12		nfcsb1v 3853	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
13	12	nfcri 2893	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
14		nfcsb1v 3853	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfeq2 2923	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
17	5	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
18	4, 17	anbi12d 630	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)))
19	11, 16, 18	cbvexv1 2341	. . . . . 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 2742	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
22	21	anbi2d 628	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
23	22	exbidv 1925	. . . . 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 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 5853	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
27	24, 26	elab2g 3604	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
28	9, 27	syl5ibr 245	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹))
29	28	impcom 407	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064  ⦋csb 3828   ↦ cmpt 5153  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-mpt 5154  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  fliftel1  7161  minveclem4  24501  minvecolem4  29143  rexunirn  30741  esum2d  31961  exrecfnlem  35477  totbndbnd  35874  rrnequiv  35920  suprnmpt  42599  disjf1o  42618  disjinfi  42620  choicefi  42629  elrnmpt1d  42662  suprubrnmpt  42688  supxrleubrnmpt  42836  suprleubrnmpt  42852  infrnmptle  42853  infxrunb3rnmpt  42858  supminfrnmpt  42875  infxrgelbrnmpt  42884  fourierdlem31  43569  ioorrnopnlem  43735  sge0f1o  43810  sge0supre  43817  sge0gerp  43823  sge0iunmpt  43846  sge0rernmpt  43850  sge0reuz  43875  meadjiunlem  43893  iunhoiioolem  44103  vonioolem1  44108  smfpimcclem  44227