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Theorem elrnmpt1 5911
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 3447 . . . 4 𝑥 ∈ V
2 id 22 . . . . . . 7 (𝑥 = 𝑧𝑥 = 𝑧)
3 csbeq1a 3867 . . . . . . 7 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
42, 3eleq12d 2832 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
5 csbeq1a 3867 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
65biantrud 532 . . . . . 6 (𝑥 = 𝑧 → (𝑧𝑧 / 𝑥𝐴 ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
74, 6bitr2d 279 . . . . 5 (𝑥 = 𝑧 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
87equcoms 2023 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
91, 8spcev 3563 . . 3 (𝑥𝐴 → ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵))
10 df-rex 3072 . . . . . 6 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
11 nfv 1917 . . . . . . 7 𝑧(𝑥𝐴𝑦 = 𝐵)
12 nfcsb1v 3878 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
1312nfcri 2892 . . . . . . . 8 𝑥 𝑧𝑧 / 𝑥𝐴
14 nfcsb1v 3878 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2922 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐵
1613, 15nfan 1902 . . . . . . 7 𝑥(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)
175eqeq2d 2747 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
184, 17anbi12d 631 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)))
1911, 16, 18cbvexv1 2338 . . . . . 6 (∃𝑥(𝑥𝐴𝑦 = 𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
2010, 19bitri 274 . . . . 5 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
21 eqeq1 2740 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
2221anbi2d 629 . . . . . 6 (𝑦 = 𝐵 → ((𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2322exbidv 1924 . . . . 5 (𝑦 = 𝐵 → (∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2420, 23bitrid 282 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
25 rnmpt.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
2625rnmpt 5908 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
2724, 26elab2g 3630 . . 3 (𝐵𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
289, 27syl5ibr 245 . 2 (𝐵𝑉 → (𝑥𝐴𝐵 ∈ ran 𝐹))
2928impcom 408 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3071  csb 3853  cmpt 5186  ran crn 5632
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-mpt 5187  df-cnv 5639  df-dm 5641  df-rn 5642
This theorem is referenced by:  fliftel1  7251  minveclem4  24780  minvecolem4  29708  rexunirn  31306  esum2d  32561  exrecfnlem  35817  totbndbnd  36215  rrnequiv  36261  suprnmpt  43327  disjf1o  43346  disjinfi  43348  choicefi  43357  elrnmpt1d  43391  suprubrnmpt  43417  supxrleubrnmpt  43577  suprleubrnmpt  43593  infrnmptle  43594  infxrunb3rnmpt  43599  supminfrnmpt  43616  infxrgelbrnmpt  43625  fourierdlem31  44311  ioorrnopnlem  44477  sge0f1o  44555  sge0supre  44562  sge0gerp  44568  sge0iunmpt  44591  sge0rernmpt  44595  sge0reuz  44620  meadjiunlem  44638  iunhoiioolem  44848  vonioolem1  44853  smfpimcclem  44980
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