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Theorem elrnmpt 4610
Description: The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
Hypothesis
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
elrnmpt (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2062 . . 3 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
21rexbidv 2344 . 2 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
3 rnmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43rnmpt 4609 . 2 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
52, 4elab2g 2711 1 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  wrex 2324  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-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:  elrnmpt1s  4611
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