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Theorem elrnmpt1 5971
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 3484 . . . 4 𝑥 ∈ V
2 id 22 . . . . . . 7 (𝑥 = 𝑧𝑥 = 𝑧)
3 csbeq1a 3913 . . . . . . 7 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
42, 3eleq12d 2835 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
5 csbeq1a 3913 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
65biantrud 531 . . . . . 6 (𝑥 = 𝑧 → (𝑧𝑧 / 𝑥𝐴 ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
74, 6bitr2d 280 . . . . 5 (𝑥 = 𝑧 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
87equcoms 2019 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
91, 8spcev 3606 . . 3 (𝑥𝐴 → ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵))
10 df-rex 3071 . . . . . 6 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
11 nfv 1914 . . . . . . 7 𝑧(𝑥𝐴𝑦 = 𝐵)
12 nfcsb1v 3923 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
1312nfcri 2897 . . . . . . . 8 𝑥 𝑧𝑧 / 𝑥𝐴
14 nfcsb1v 3923 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2923 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐵
1613, 15nfan 1899 . . . . . . 7 𝑥(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)
175eqeq2d 2748 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
184, 17anbi12d 632 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)))
1911, 16, 18cbvexv1 2344 . . . . . 6 (∃𝑥(𝑥𝐴𝑦 = 𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
2010, 19bitri 275 . . . . 5 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
21 eqeq1 2741 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
2221anbi2d 630 . . . . . 6 (𝑦 = 𝐵 → ((𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2322exbidv 1921 . . . . 5 (𝑦 = 𝐵 → (∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2420, 23bitrid 283 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
25 rnmpt.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
2625rnmpt 5968 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
2724, 26elab2g 3680 . . 3 (𝐵𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
289, 27imbitrrid 246 . 2 (𝐵𝑉 → (𝑥𝐴𝐵 ∈ ran 𝐹))
2928impcom 407 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  csb 3899  cmpt 5225  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  elrnmpt1d  5975  fliftel1  7330  minveclem4  25466  minvecolem4  30899  rexunirn  32511  esum2d  34094  exrecfnlem  37380  totbndbnd  37796  rrnequiv  37842  suprnmpt  45179  disjf1o  45196  disjinfi  45197  choicefi  45205  suprubrnmpt  45260  supxrleubrnmpt  45417  suprleubrnmpt  45433  infrnmptle  45434  infxrunb3rnmpt  45439  supminfrnmpt  45456  infxrgelbrnmpt  45465  fourierdlem31  46153  ioorrnopnlem  46319  sge0supre  46404  sge0gerp  46410  sge0iunmpt  46433  sge0rernmpt  46437  sge0reuz  46462  meadjiunlem  46480  iunhoiioolem  46690  vonioolem1  46695  smfpimcclem  46822
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