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Theorem dfmpt3 4947
Description: Alternate definition for the "maps to" notation df-mpt 3794. (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 3794 . 2 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
2 elsn 3365 . . . . . . 7 (y {B} ↔ y = B)
32anbi2i 433 . . . . . 6 ((x A y {B}) ↔ (x A y = B))
43anbi2i 433 . . . . 5 ((z = ⟨x, y (x A y {B})) ↔ (z = ⟨x, y (x A y = B)))
542exbii 1479 . . . 4 (xy(z = ⟨x, y (x A y {B})) ↔ xy(z = ⟨x, y (x A y = B)))
6 eliunxp 4402 . . . 4 (z x A ({x} × {B}) ↔ xy(z = ⟨x, y (x A y {B})))
7 elopab 3969 . . . 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 2019 . 2 x A ({x} × {B}) = {⟨x, y⟩ ∣ (x A y = B)}
101, 9eqtr4i 2045 1 (x AB) = x A ({x} × {B})
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374  {csn 3350  cop 3353   ciun 3631  {copab 3791  cmpt 3792   × cxp 4270
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-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-opab 3793  df-mpt 3794  df-xp 4278  df-rel 4279
This theorem is referenced by:  dfmpt  5265  dfmptg  5267
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