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Theorem elrnmpt1s 4527
 Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
rnmpt.1 𝐹 = (x AB)
elrnmpt1s.1 (x = 𝐷B = 𝐶)
Assertion
Ref Expression
elrnmpt1s ((𝐷 A 𝐶 𝑉) → 𝐶 ran 𝐹)
Distinct variable groups:   x,𝐶   x,A   x,𝐷
Allowed substitution hints:   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem elrnmpt1s
StepHypRef Expression
1 eqid 2037 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . . 5 (x = 𝐷B = 𝐶)
32eqeq2d 2048 . . . 4 (x = 𝐷 → (𝐶 = B𝐶 = 𝐶))
43rspcev 2650 . . 3 ((𝐷 A 𝐶 = 𝐶) → x A 𝐶 = B)
51, 4mpan2 401 . 2 (𝐷 Ax A 𝐶 = B)
6 rnmpt.1 . . . 4 𝐹 = (x AB)
76elrnmpt 4526 . . 3 (𝐶 𝑉 → (𝐶 ran 𝐹x A 𝐶 = B))
87biimparc 283 . 2 ((x A 𝐶 = B 𝐶 𝑉) → 𝐶 ran 𝐹)
95, 8sylan 267 1 ((𝐷 A 𝐶 𝑉) → 𝐶 ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ∃wrex 2301   ↦ cmpt 3809  ran crn 4289 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by: (None)
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