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Theorem repizf2lem 3905
Description: Lemma for repizf2 3906. If we have a function-like proposition which provides at most one value of y for each x in a set w, we can change "at most one" to "exactly one" by restricting the values of x to those values for which the proposition provides a value of y. (Contributed by Jim Kingdon, 7-Sep-2018.)
Assertion
Ref Expression
repizf2lem (x w ∃*yφx {x wyφ}∃!yφ)

Proof of Theorem repizf2lem
StepHypRef Expression
1 df-mo 1901 . . . 4 (∃*yφ ↔ (yφ∃!yφ))
21imbi2i 215 . . 3 ((x w∃*yφ) ↔ (x w → (yφ∃!yφ)))
32albii 1356 . 2 (x(x w∃*yφ) ↔ x(x w → (yφ∃!yφ)))
4 df-ral 2305 . 2 (x w ∃*yφx(x w∃*yφ))
5 df-ral 2305 . . 3 (x {x wyφ}∃!yφx(x {x wyφ} → ∃!yφ))
6 rabid 2479 . . . . . 6 (x {x wyφ} ↔ (x w yφ))
76imbi1i 227 . . . . 5 ((x {x wyφ} → ∃!yφ) ↔ ((x w yφ) → ∃!yφ))
8 impexp 250 . . . . 5 (((x w yφ) → ∃!yφ) ↔ (x w → (yφ∃!yφ)))
97, 8bitri 173 . . . 4 ((x {x wyφ} → ∃!yφ) ↔ (x w → (yφ∃!yφ)))
109albii 1356 . . 3 (x(x {x wyφ} → ∃!yφ) ↔ x(x w → (yφ∃!yφ)))
115, 10bitri 173 . 2 (x {x wyφ}∃!yφx(x w → (yφ∃!yφ)))
123, 4, 113bitr4i 201 1 (x w ∃*yφx {x wyφ}∃!yφ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378   wcel 1390  ∃!weu 1897  ∃*wmo 1898  wral 2300  {crab 2304
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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-sb 1643  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rab 2309
This theorem is referenced by:  repizf2  3906
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