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Theorem elrnmpt1s 4612
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 2056 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
32eqeq2d 2067 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
43rspcev 2673 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
51, 4mpan2 409 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
6 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
76elrnmpt 4611 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
87biimparc 287 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
95, 8sylan 271 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  wrex 2324  cmpt 3846  ran crn 4374
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 3903  ax-pow 3955  ax-pr 3972
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-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by: (None)
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