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Theorem elrnmpt1 5806
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 3476 . . . 4 𝑥 ∈ V
2 id 22 . . . . . . 7 (𝑥 = 𝑧𝑥 = 𝑧)
3 csbeq1a 3874 . . . . . . 7 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
42, 3eleq12d 2905 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
5 csbeq1a 3874 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
65biantrud 534 . . . . . 6 (𝑥 = 𝑧 → (𝑧𝑧 / 𝑥𝐴 ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
74, 6bitr2d 282 . . . . 5 (𝑥 = 𝑧 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
87equcoms 2027 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
91, 8spcev 3586 . . 3 (𝑥𝐴 → ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵))
10 df-rex 3131 . . . . . 6 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
11 nfv 1915 . . . . . . 7 𝑧(𝑥𝐴𝑦 = 𝐵)
12 nfcsb1v 3884 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
1312nfcri 2967 . . . . . . . 8 𝑥 𝑧𝑧 / 𝑥𝐴
14 nfcsb1v 3884 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2990 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐵
1613, 15nfan 1900 . . . . . . 7 𝑥(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)
175eqeq2d 2831 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
184, 17anbi12d 632 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)))
1911, 16, 18cbvexv1 2362 . . . . . 6 (∃𝑥(𝑥𝐴𝑦 = 𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
2010, 19bitri 277 . . . . 5 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
21 eqeq1 2824 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
2221anbi2d 630 . . . . . 6 (𝑦 = 𝐵 → ((𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2322exbidv 1922 . . . . 5 (𝑦 = 𝐵 → (∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2420, 23syl5bb 285 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
25 rnmpt.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
2625rnmpt 5803 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
2724, 26elab2g 3648 . . 3 (𝐵𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
289, 27syl5ibr 248 . 2 (𝐵𝑉 → (𝑥𝐴𝐵 ∈ ran 𝐹))
2928impcom 410 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wex 1780  wcel 2114  wrex 3126  csb 3860  cmpt 5122  ran crn 5532
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 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
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 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-mpt 5123  df-cnv 5539  df-dm 5541  df-rn 5542
This theorem is referenced by:  fliftel1  7040  minveclem4  24015  minvecolem4  28642  rexunirn  30241  esum2d  31360  exrecfnlem  34677  totbndbnd  35103  rrnequiv  35149  suprnmpt  41584  disjf1o  41606  disjinfi  41608  choicefi  41617  elrnmpt1d  41654  suprubrnmpt  41679  supxrleubrnmpt  41833  suprleubrnmpt  41850  infrnmptle  41851  infxrunb3rnmpt  41856  supminfrnmpt  41873  infxrgelbrnmpt  41884  fourierdlem31  42571  ioorrnopnlem  42737  sge0f1o  42812  sge0supre  42819  sge0gerp  42825  sge0iunmpt  42848  sge0rernmpt  42852  sge0reuz  42877  meadjiunlem  42895  iunhoiioolem  43105  vonioolem1  43110  smfpimcclem  43229
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