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Theorem elimasng 4636
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
Assertion
Ref Expression
elimasng ((B 𝑉 𝐶 𝑊) → (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A))

Proof of Theorem elimasng
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 3378 . . . . 5 (y = B → {y} = {B})
21imaeq2d 4611 . . . 4 (y = B → (A “ {y}) = (A “ {B}))
32eleq2d 2104 . . 3 (y = B → (z (A “ {y}) ↔ z (A “ {B})))
4 opeq1 3540 . . . 4 (y = B → ⟨y, z⟩ = ⟨B, z⟩)
54eleq1d 2103 . . 3 (y = B → (⟨y, z A ↔ ⟨B, z A))
63, 5bibi12d 224 . 2 (y = B → ((z (A “ {y}) ↔ ⟨y, z A) ↔ (z (A “ {B}) ↔ ⟨B, z A)))
7 eleq1 2097 . . 3 (z = 𝐶 → (z (A “ {B}) ↔ 𝐶 (A “ {B})))
8 opeq2 3541 . . . 4 (z = 𝐶 → ⟨B, z⟩ = ⟨B, 𝐶⟩)
98eleq1d 2103 . . 3 (z = 𝐶 → (⟨B, z A ↔ ⟨B, 𝐶 A))
107, 9bibi12d 224 . 2 (z = 𝐶 → ((z (A “ {B}) ↔ ⟨B, z A) ↔ (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A)))
11 vex 2554 . . 3 y V
12 vex 2554 . . 3 z V
1311, 12elimasn 4635 . 2 (z (A “ {y}) ↔ ⟨y, z A)
146, 10, 13vtocl2g 2611 1 ((B 𝑉 𝐶 𝑊) → (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {csn 3367  cop 3370  cima 4291
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-sbc 2759  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-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  eliniseg  4638  inimasn  4684  dffv3g  5117  fvimacnv  5225  funfvima3  5335  elecg  6080
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