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Theorem elrnmpt1s 4835
 Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
elrnmpt1s.1 (𝑥 = 𝐷𝐵 = 𝐶)
Assertion
Ref Expression
elrnmpt1s ((𝐷𝐴𝐶𝑉) → 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 2157 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
32eqeq2d 2169 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
43rspcev 2816 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
51, 4mpan2 422 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
6 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
76elrnmpt 4834 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
87biimparc 297 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
95, 8sylan 281 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  ∃wrex 2436   ↦ cmpt 4025  ran crn 4586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-mpt 4027  df-cnv 4593  df-dm 4595  df-rn 4596 This theorem is referenced by: (None)
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