Theorem elrnmpt1 5957

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 3478	. . . 4 ⊢ 𝑥 ∈ V
2		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3		csbeq1a 3907	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
4	2, 3	eleq12d 2827	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
5		csbeq1a 3907	. . . . . . 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 3596	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10		df-rex 3071	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
11		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)
12		nfcsb1v 3918	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
13	12	nfcri 2890	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
14		nfcsb1v 3918	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfeq2 2920	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
16	13, 15	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
17	5	eqeq2d 2743	. . . . . . . 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 2736	. . . . . . 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 5954	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
27	24, 26	elab2g 3670	. . 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 3070  ⦋csb 3893   ↦ cmpt 5231  ran crn 5677

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687

This theorem is referenced by:  fliftel1  7309  minveclem4  24956  minvecolem4  30171  rexunirn  31770  esum2d  33160  exrecfnlem  36346  totbndbnd  36743  rrnequiv  36789  suprnmpt  43952  disjf1o  43969  disjinfi  43970  choicefi  43978  elrnmpt1d  44011  suprubrnmpt  44036  supxrleubrnmpt  44195  suprleubrnmpt  44211  infrnmptle  44212  infxrunb3rnmpt  44217  supminfrnmpt  44234  infxrgelbrnmpt  44243  fourierdlem31  44933  ioorrnopnlem  45099  sge0f1o  45177  sge0supre  45184  sge0gerp  45190  sge0iunmpt  45213  sge0rernmpt  45217  sge0reuz  45242  meadjiunlem  45260  iunhoiioolem  45470  vonioolem1  45475  smfpimcclem  45602