Theorem elrnmpt1 5830

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 3497	. . . 4 ⊢ 𝑥 ∈ V
2		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3		csbeq1a 3897	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
4	2, 3	eleq12d 2907	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
5		csbeq1a 3897	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
6	5	biantrud 534	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
7	4, 6	bitr2d 282	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ↔ 𝑥 ∈ 𝐴))
8	7	equcoms 2027	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) ↔ 𝑥 ∈ 𝐴))
9	1, 8	spcev 3607	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10		df-rex 3144	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
11		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)
12		nfcsb1v 3907	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
13	12	nfcri 2971	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
14		nfcsb1v 3907	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfeq2 2995	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
17	5	eqeq2d 2832	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
18	4, 17	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)))
19	11, 16, 18	cbvexv1 2362	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
20	10, 19	bitri 277	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
21		eqeq1 2825	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
22	21	anbi2d 630	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
23	22	exbidv 1922	. . . . 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 285	. . . 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 5827	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
27	24, 26	elab2g 3668	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))
28	9, 27	syl5ibr 248	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹))
29	28	impcom 410	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃wrex 3139  ⦋csb 3883   ↦ cmpt 5146  ran crn 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-mpt 5147  df-cnv 5563  df-dm 5565  df-rn 5566

This theorem is referenced by:  fliftel1  7063  minveclem4  24035  minvecolem4  28657  rexunirn  30256  esum2d  31352  exrecfnlem  34663  totbndbnd  35082  rrnequiv  35128  suprnmpt  41450  disjf1o  41472  disjinfi  41474  choicefi  41483  elrnmpt1d  41520  suprubrnmpt  41545  supxrleubrnmpt  41699  suprleubrnmpt  41716  infrnmptle  41717  infxrunb3rnmpt  41722  supminfrnmpt  41739  infxrgelbrnmpt  41750  fourierdlem31  42443  ioorrnopnlem  42609  sge0f1o  42684  sge0supre  42691  sge0gerp  42697  sge0iunmpt  42720  sge0rernmpt  42724  sge0reuz  42749  meadjiunlem  42767  iunhoiioolem  42977  vonioolem1  42982  smfpimcclem  43101