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Theorem fv3 5312
Description: Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fv3 (𝐹𝐴) = {𝑥 ∣ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐴,𝑦

Proof of Theorem fv3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elfv 5287 . . 3 (𝑥 ∈ (𝐹𝐴) ↔ ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
2 bi2 128 . . . . . . . . . 10 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝑦 = 𝑧𝐴𝐹𝑦))
32alimi 1389 . . . . . . . . 9 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧𝐴𝐹𝑦))
4 vex 2622 . . . . . . . . . 10 𝑧 ∈ V
5 breq2 3841 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝐴𝐹𝑦𝐴𝐹𝑧))
64, 5ceqsalv 2649 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑧𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧)
73, 6sylib 120 . . . . . . . 8 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → 𝐴𝐹𝑧)
87anim2i 334 . . . . . . 7 ((𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → (𝑥𝑧𝐴𝐹𝑧))
98eximi 1536 . . . . . 6 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑧(𝑥𝑧𝐴𝐹𝑧))
10 elequ2 1648 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
11 breq2 3841 . . . . . . . 8 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
1210, 11anbi12d 457 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥𝑧𝐴𝐹𝑧) ↔ (𝑥𝑦𝐴𝐹𝑦)))
1312cbvexv 1843 . . . . . 6 (∃𝑧(𝑥𝑧𝐴𝐹𝑧) ↔ ∃𝑦(𝑥𝑦𝐴𝐹𝑦))
149, 13sylib 120 . . . . 5 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑦(𝑥𝑦𝐴𝐹𝑦))
15 exsimpr 1554 . . . . . 6 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
16 df-eu 1951 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
1715, 16sylibr 132 . . . . 5 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦)
1814, 17jca 300 . . . 4 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
19 nfeu1 1959 . . . . . . 7 𝑦∃!𝑦 𝐴𝐹𝑦
20 nfv 1466 . . . . . . . . 9 𝑦 𝑥𝑧
21 nfa1 1479 . . . . . . . . 9 𝑦𝑦(𝐴𝐹𝑦𝑦 = 𝑧)
2220, 21nfan 1502 . . . . . . . 8 𝑦(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
2322nfex 1573 . . . . . . 7 𝑦𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
2419, 23nfim 1509 . . . . . 6 𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
25 bi1 116 . . . . . . . . . . . . . 14 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝐴𝐹𝑦𝑦 = 𝑧))
26 ax-14 1450 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
2725, 26syl6 33 . . . . . . . . . . . . 13 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝐴𝐹𝑦 → (𝑥𝑦𝑥𝑧)))
2827com23 77 . . . . . . . . . . . 12 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝑥𝑦 → (𝐴𝐹𝑦𝑥𝑧)))
2928impd 251 . . . . . . . . . . 11 ((𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → 𝑥𝑧))
3029sps 1475 . . . . . . . . . 10 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → 𝑥𝑧))
3130anc2ri 323 . . . . . . . . 9 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → (𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3231com12 30 . . . . . . . 8 ((𝑥𝑦𝐴𝐹𝑦) → (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → (𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3332eximdv 1808 . . . . . . 7 ((𝑥𝑦𝐴𝐹𝑦) → (∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3416, 33syl5bi 150 . . . . . 6 ((𝑥𝑦𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3524, 34exlimi 1530 . . . . 5 (∃𝑦(𝑥𝑦𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3635imp 122 . . . 4 ((∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
3718, 36impbii 124 . . 3 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) ↔ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
381, 37bitri 182 . 2 (𝑥 ∈ (𝐹𝐴) ↔ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
3938abbi2i 2202 1 (𝐹𝐴) = {𝑥 ∣ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948  {cab 2074   class class class wbr 3837  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010
This theorem is referenced by: (None)
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