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Theorem elrnmptg 4513
 Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1 𝐹 = (x AB)
Assertion
Ref Expression
elrnmptg (x A B 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 elrnmptg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 rnmpt.1 . . . 4 𝐹 = (x AB)
21rnmpt 4509 . . 3 ran 𝐹 = {yx A y = B}
32eleq2i 2086 . 2 (𝐶 ran 𝐹𝐶 {yx A y = B})
4 r19.29 2428 . . . . 5 ((x A B 𝑉 x A 𝐶 = B) → x A (B 𝑉 𝐶 = B))
5 eleq1 2082 . . . . . . . 8 (𝐶 = B → (𝐶 𝑉B 𝑉))
65biimparc 283 . . . . . . 7 ((B 𝑉 𝐶 = B) → 𝐶 𝑉)
7 elex 2543 . . . . . . 7 (𝐶 𝑉𝐶 V)
86, 7syl 14 . . . . . 6 ((B 𝑉 𝐶 = B) → 𝐶 V)
98rexlimivw 2407 . . . . 5 (x A (B 𝑉 𝐶 = B) → 𝐶 V)
104, 9syl 14 . . . 4 ((x A B 𝑉 x A 𝐶 = B) → 𝐶 V)
1110ex 108 . . 3 (x A B 𝑉 → (x A 𝐶 = B𝐶 V))
12 eqeq1 2028 . . . . 5 (y = 𝐶 → (y = B𝐶 = B))
1312rexbidv 2305 . . . 4 (y = 𝐶 → (x A y = Bx A 𝐶 = B))
1413elab3g 2670 . . 3 ((x A 𝐶 = B𝐶 V) → (𝐶 {yx A y = B} ↔ x A 𝐶 = B))
1511, 14syl 14 . 2 (x A B 𝑉 → (𝐶 {yx A y = B} ↔ x A 𝐶 = B))
163, 15syl5bb 181 1 (x A B 𝑉 → (𝐶 ran 𝐹x A 𝐶 = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  {cab 2008  ∀wral 2284  ∃wrex 2285  Vcvv 2535   ↦ cmpt 3792  ran crn 4273 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-mpt 3794  df-cnv 4280  df-dm 4282  df-rn 4283 This theorem is referenced by:  elrnmpti  4514  fliftel  5358
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