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Theorem elrnmpt 4506
 Description: The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
Hypothesis
Ref Expression
rnmpt.1 𝐹 = (x AB)
Assertion
Ref Expression
elrnmpt (𝐶 𝑉 → (𝐶 ran 𝐹x A 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 2024 . . 3 (y = 𝐶 → (y = B𝐶 = B))
21rexbidv 2301 . 2 (y = 𝐶 → (x A y = Bx A 𝐶 = B))
3 rnmpt.1 . . 3 𝐹 = (x AB)
43rnmpt 4505 . 2 ran 𝐹 = {yx A y = B}
52, 4elab2g 2662 1 (𝐶 𝑉 → (𝐶 ran 𝐹x A 𝐶 = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226   ∈ wcel 1370  ∃wrex 2281   ↦ cmpt 3788  ran crn 4269 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-mpt 3790  df-cnv 4276  df-dm 4278  df-rn 4279 This theorem is referenced by:  elrnmpt1s  4507
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