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Theorem fv3 6355
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 6338 . . 3 (𝑥 ∈ (𝐹𝐴) ↔ ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
2 biimpr 210 . . . . . . . . . 10 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝑦 = 𝑧𝐴𝐹𝑦))
32alimi 1876 . . . . . . . . 9 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧𝐴𝐹𝑦))
4 vex 3331 . . . . . . . . . 10 𝑧 ∈ V
5 breq2 4796 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝐴𝐹𝑦𝐴𝐹𝑧))
64, 5ceqsalv 3361 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑧𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧)
73, 6sylib 208 . . . . . . . 8 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → 𝐴𝐹𝑧)
87anim2i 594 . . . . . . 7 ((𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → (𝑥𝑧𝐴𝐹𝑧))
98eximi 1899 . . . . . 6 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑧(𝑥𝑧𝐴𝐹𝑧))
10 elequ2 2141 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
11 breq2 4796 . . . . . . . 8 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
1210, 11anbi12d 749 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥𝑧𝐴𝐹𝑧) ↔ (𝑥𝑦𝐴𝐹𝑦)))
1312cbvexv 2408 . . . . . 6 (∃𝑧(𝑥𝑧𝐴𝐹𝑧) ↔ ∃𝑦(𝑥𝑦𝐴𝐹𝑦))
149, 13sylib 208 . . . . 5 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑦(𝑥𝑦𝐴𝐹𝑦))
15 exsimpr 1933 . . . . . 6 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
16 df-eu 2599 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
1715, 16sylibr 224 . . . . 5 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦)
1814, 17jca 555 . . . 4 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
19 nfeu1 2605 . . . . . . 7 𝑦∃!𝑦 𝐴𝐹𝑦
20 nfv 1980 . . . . . . . . 9 𝑦 𝑥𝑧
21 nfa1 2165 . . . . . . . . 9 𝑦𝑦(𝐴𝐹𝑦𝑦 = 𝑧)
2220, 21nfan 1965 . . . . . . . 8 𝑦(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
2322nfex 2289 . . . . . . 7 𝑦𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
2419, 23nfim 1962 . . . . . 6 𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
25 biimp 205 . . . . . . . . . . . . . 14 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝐴𝐹𝑦𝑦 = 𝑧))
26 ax9 2140 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
2725, 26syl6 35 . . . . . . . . . . . . 13 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝐴𝐹𝑦 → (𝑥𝑦𝑥𝑧)))
2827com23 86 . . . . . . . . . . . 12 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝑥𝑦 → (𝐴𝐹𝑦𝑥𝑧)))
2928impd 446 . . . . . . . . . . 11 ((𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → 𝑥𝑧))
3029sps 2190 . . . . . . . . . 10 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → 𝑥𝑧))
3130anc2ri 582 . . . . . . . . 9 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → (𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3231com12 32 . . . . . . . 8 ((𝑥𝑦𝐴𝐹𝑦) → (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → (𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3332eximdv 1983 . . . . . . 7 ((𝑥𝑦𝐴𝐹𝑦) → (∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3416, 33syl5bi 232 . . . . . 6 ((𝑥𝑦𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3524, 34exlimi 2221 . . . . 5 (∃𝑦(𝑥𝑦𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3635imp 444 . . . 4 ((∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
3718, 36impbii 199 . . 3 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) ↔ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
381, 37bitri 264 . 2 (𝑥 ∈ (𝐹𝐴) ↔ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
3938abbi2i 2864 1 (𝐹𝐴) = {𝑥 ∣ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1618   = wceq 1620  wex 1841  wcel 2127  ∃!weu 2595  {cab 2734   class class class wbr 4792  cfv 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-iota 6000  df-fv 6045
This theorem is referenced by: (None)
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