ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfmpt3 Structured version   GIF version

Theorem dfmpt3 4964
Description: Alternate definition for the "maps to" notation df-mpt 3811. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
dfmpt3 (x AB) = x A ({x} × {B})

Proof of Theorem dfmpt3
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt 3811 . 2 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
2 elsn 3382 . . . . . . 7 (y {B} ↔ y = B)
32anbi2i 430 . . . . . 6 ((x A y {B}) ↔ (x A y = B))
43anbi2i 430 . . . . 5 ((z = ⟨x, y (x A y {B})) ↔ (z = ⟨x, y (x A y = B)))
542exbii 1494 . . . 4 (xy(z = ⟨x, y (x A y {B})) ↔ xy(z = ⟨x, y (x A y = B)))
6 eliunxp 4418 . . . 4 (z x A ({x} × {B}) ↔ xy(z = ⟨x, y (x A y {B})))
7 elopab 3986 . . . 4 (z {⟨x, y⟩ ∣ (x A y = B)} ↔ xy(z = ⟨x, y (x A y = B)))
85, 6, 73bitr4i 201 . . 3 (z x A ({x} × {B}) ↔ z {⟨x, y⟩ ∣ (x A y = B)})
98eqriv 2034 . 2 x A ({x} × {B}) = {⟨x, y⟩ ∣ (x A y = B)}
101, 9eqtr4i 2060 1 (x AB) = x A ({x} × {B})
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {csn 3367  cop 3370   ciun 3648  {copab 3808  cmpt 3809   × cxp 4286
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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  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-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295
This theorem is referenced by:  dfmpt  5283  dfmptg  5285
  Copyright terms: Public domain W3C validator