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Theorem fnasrng 5286
 Description: A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)
Assertion
Ref Expression
fnasrng (x A B 𝑉 → (x AB) = ran (x A ↦ ⟨x, B⟩))

Proof of Theorem fnasrng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfmptg 5285 . 2 (x A B 𝑉 → (x AB) = x A {⟨x, B⟩})
2 eqid 2037 . . . . 5 (x A ↦ ⟨x, B⟩) = (x A ↦ ⟨x, B⟩)
32rnmpt 4525 . . . 4 ran (x A ↦ ⟨x, B⟩) = {yx A y = ⟨x, B⟩}
4 elsn 3382 . . . . . 6 (y {⟨x, B⟩} ↔ y = ⟨x, B⟩)
54rexbii 2325 . . . . 5 (x A y {⟨x, B⟩} ↔ x A y = ⟨x, B⟩)
65abbii 2150 . . . 4 {yx A y {⟨x, B⟩}} = {yx A y = ⟨x, B⟩}
73, 6eqtr4i 2060 . . 3 ran (x A ↦ ⟨x, B⟩) = {yx A y {⟨x, B⟩}}
8 df-iun 3650 . . 3 x A {⟨x, B⟩} = {yx A y {⟨x, B⟩}}
97, 8eqtr4i 2060 . 2 ran (x A ↦ ⟨x, B⟩) = x A {⟨x, B⟩}
101, 9syl6eqr 2087 1 (x A B 𝑉 → (x AB) = ran (x A ↦ ⟨x, B⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  {csn 3367  ⟨cop 3370  ∪ ciun 3648   ↦ 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-reu 2307  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852 This theorem is referenced by:  resfunexg  5325
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