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Theorem elrnmpt1 5295
 Description: Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
elrnmpt1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)

Proof of Theorem elrnmpt1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . 4 𝑥 ∈ V
2 id 22 . . . . . . 7 (𝑥 = 𝑧𝑥 = 𝑧)
3 csbeq1a 3508 . . . . . . 7 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
42, 3eleq12d 2682 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
5 csbeq1a 3508 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
65biantrud 527 . . . . . 6 (𝑥 = 𝑧 → (𝑧𝑧 / 𝑥𝐴 ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
74, 6bitr2d 268 . . . . 5 (𝑥 = 𝑧 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
87equcoms 1934 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
91, 8spcev 3273 . . 3 (𝑥𝐴 → ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵))
10 df-rex 2902 . . . . . 6 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
11 nfv 1830 . . . . . . 7 𝑧(𝑥𝐴𝑦 = 𝐵)
12 nfcsb1v 3515 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
1312nfcri 2745 . . . . . . . 8 𝑥 𝑧𝑧 / 𝑥𝐴
14 nfcsb1v 3515 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2766 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐵
1613, 15nfan 1816 . . . . . . 7 𝑥(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)
175eqeq2d 2620 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
184, 17anbi12d 743 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)))
1911, 16, 18cbvex 2260 . . . . . 6 (∃𝑥(𝑥𝐴𝑦 = 𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
2010, 19bitri 263 . . . . 5 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
21 eqeq1 2614 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
2221anbi2d 736 . . . . . 6 (𝑦 = 𝐵 → ((𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2322exbidv 1837 . . . . 5 (𝑦 = 𝐵 → (∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2420, 23syl5bb 271 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
25 rnmpt.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
2625rnmpt 5292 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
2724, 26elab2g 3322 . . 3 (𝐵𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
289, 27syl5ibr 235 . 2 (𝐵𝑉 → (𝑥𝐴𝐵 ∈ ran 𝐹))
2928impcom 445 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  ⦋csb 3499   ↦ cmpt 4643  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  fliftel1  6460  minveclem4  23011  minvecolem4  27120  rexunirn  28715  esum2d  29482  totbndbnd  32758  rrnequiv  32804  suprnmpt  38350  disjf1o  38373  disjinfi  38375  choicefi  38387  fourierdlem31  39031  ioorrnopnlem  39200  sge0f1o  39275  sge0supre  39282  sge0gerp  39288  sge0iunmpt  39311  sge0rernmpt  39315  sge0reuz  39340  meadjiunlem  39358  iunhoiioolem  39566  vonioolem1  39571
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