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Theorem elrnmpt1 4612
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 2577 . . . 4 𝑥 ∈ V
2 id 19 . . . . . . 7 (𝑥 = 𝑧𝑥 = 𝑧)
3 csbeq1a 2887 . . . . . . 7 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
42, 3eleq12d 2124 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
5 csbeq1a 2887 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
65biantrud 292 . . . . . 6 (𝑥 = 𝑧 → (𝑧𝑧 / 𝑥𝐴 ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
74, 6bitr2d 182 . . . . 5 (𝑥 = 𝑧 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
87equcoms 1610 . . . 4 (𝑧 = 𝑥 → ((𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵) ↔ 𝑥𝐴))
91, 8spcev 2664 . . 3 (𝑥𝐴 → ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵))
10 df-rex 2329 . . . . . 6 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
11 nfv 1437 . . . . . . 7 𝑧(𝑥𝐴𝑦 = 𝐵)
12 nfcsb1v 2909 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐴
1312nfcri 2188 . . . . . . . 8 𝑥 𝑧𝑧 / 𝑥𝐴
14 nfcsb1v 2909 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2205 . . . . . . . 8 𝑥 𝑦 = 𝑧 / 𝑥𝐵
1613, 15nfan 1473 . . . . . . 7 𝑥(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)
175eqeq2d 2067 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
184, 17anbi12d 450 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵)))
1911, 16, 18cbvex 1655 . . . . . 6 (∃𝑥(𝑥𝐴𝑦 = 𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
2010, 19bitri 177 . . . . 5 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵))
21 eqeq1 2062 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
2221anbi2d 445 . . . . . 6 (𝑦 = 𝐵 → ((𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ (𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2322exbidv 1722 . . . . 5 (𝑦 = 𝐵 → (∃𝑧(𝑧𝑧 / 𝑥𝐴𝑦 = 𝑧 / 𝑥𝐵) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
2420, 23syl5bb 185 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
25 rnmpt.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
2625rnmpt 4609 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
2724, 26elab2g 2711 . . 3 (𝐵𝑉 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴𝐵 = 𝑧 / 𝑥𝐵)))
289, 27syl5ibr 149 . 2 (𝐵𝑉 → (𝑥𝐴𝐵 ∈ ran 𝐹))
2928impcom 120 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wrex 2324  csb 2879  cmpt 3845  ran crn 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-mpt 3847  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  fliftel1  5461
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