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Theorem elrnmptg 4529
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 4525 . . 3 ran 𝐹 = {yx A y = B}
32eleq2i 2101 . 2 (𝐶 ran 𝐹𝐶 {yx A y = B})
4 r19.29 2444 . . . . 5 ((x A B 𝑉 x A 𝐶 = B) → x A (B 𝑉 𝐶 = B))
5 eleq1 2097 . . . . . . . 8 (𝐶 = B → (𝐶 𝑉B 𝑉))
65biimparc 283 . . . . . . 7 ((B 𝑉 𝐶 = B) → 𝐶 𝑉)
7 elex 2560 . . . . . . 7 (𝐶 𝑉𝐶 V)
86, 7syl 14 . . . . . 6 ((B 𝑉 𝐶 = B) → 𝐶 V)
98rexlimivw 2423 . . . . 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 2043 . . . . 5 (y = 𝐶 → (y = B𝐶 = B))
1312rexbidv 2321 . . . 4 (y = 𝐶 → (x A y = Bx A 𝐶 = B))
1413elab3g 2687 . . 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 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  cmpt 3809  ran crn 4289
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  elrnmpti  4530  fliftel  5376
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