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Theorem bdinex1 10392
Description: Bounded version of inex1 3918. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdinex1.bd BOUNDED 𝐵
bdinex1.1 𝐴 ∈ V
Assertion
Ref Expression
bdinex1 (𝐴𝐵) ∈ V

Proof of Theorem bdinex1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdinex1.1 . . . 4 𝐴 ∈ V
2 bdinex1.bd . . . . . 6 BOUNDED 𝐵
32bdeli 10339 . . . . 5 BOUNDED 𝑦𝐵
43bdzfauscl 10383 . . . 4 (𝐴 ∈ V → ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
51, 4ax-mp 7 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
6 dfcleq 2050 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
7 elin 3153 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
87bibi2i 220 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
98albii 1375 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
106, 9bitri 177 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
1110exbii 1512 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
125, 11mpbir 138 . 2 𝑥 𝑥 = (𝐴𝐵)
1312issetri 2581 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cin 2943  BOUNDED wbdc 10333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bdsep 10377
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-bdc 10334
This theorem is referenced by:  bdinex2  10393  bdinex1g  10394  bdpeano5  10441
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