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Theorem fv3 6681
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 6661 . . 3 (𝑥 ∈ (𝐹𝐴) ↔ ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
2 biimpr 221 . . . . . . . . . 10 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝑦 = 𝑧𝐴𝐹𝑦))
32alimi 1803 . . . . . . . . 9 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧𝐴𝐹𝑦))
4 breq2 5061 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝐴𝐹𝑦𝐴𝐹𝑧))
54equsalvw 2001 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑧𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧)
63, 5sylib 219 . . . . . . . 8 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → 𝐴𝐹𝑧)
76anim2i 616 . . . . . . 7 ((𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → (𝑥𝑧𝐴𝐹𝑧))
87eximi 1826 . . . . . 6 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑧(𝑥𝑧𝐴𝐹𝑧))
9 elequ2 2120 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
10 breq2 5061 . . . . . . . 8 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
119, 10anbi12d 630 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥𝑧𝐴𝐹𝑧) ↔ (𝑥𝑦𝐴𝐹𝑦)))
1211cbvexvw 2035 . . . . . 6 (∃𝑧(𝑥𝑧𝐴𝐹𝑧) ↔ ∃𝑦(𝑥𝑦𝐴𝐹𝑦))
138, 12sylib 219 . . . . 5 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑦(𝑥𝑦𝐴𝐹𝑦))
14 exsimpr 1861 . . . . . 6 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
15 eu6 2652 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
1614, 15sylibr 235 . . . . 5 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦)
1713, 16jca 512 . . . 4 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) → (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
18 nfeu1 2667 . . . . . . 7 𝑦∃!𝑦 𝐴𝐹𝑦
19 nfv 1906 . . . . . . . . 9 𝑦 𝑥𝑧
20 nfa1 2146 . . . . . . . . 9 𝑦𝑦(𝐴𝐹𝑦𝑦 = 𝑧)
2119, 20nfan 1891 . . . . . . . 8 𝑦(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
2221nfex 2334 . . . . . . 7 𝑦𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))
2318, 22nfim 1888 . . . . . 6 𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
24 biimp 216 . . . . . . . . . . . . 13 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝐴𝐹𝑦𝑦 = 𝑧))
25 ax9 2119 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
2624, 25syl6 35 . . . . . . . . . . . 12 ((𝐴𝐹𝑦𝑦 = 𝑧) → (𝐴𝐹𝑦 → (𝑥𝑦𝑥𝑧)))
2726impcomd 412 . . . . . . . . . . 11 ((𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → 𝑥𝑧))
2827sps 2174 . . . . . . . . . 10 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → 𝑥𝑧))
2928anc2ri 557 . . . . . . . . 9 (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ((𝑥𝑦𝐴𝐹𝑦) → (𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3029com12 32 . . . . . . . 8 ((𝑥𝑦𝐴𝐹𝑦) → (∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → (𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3130eximdv 1909 . . . . . . 7 ((𝑥𝑦𝐴𝐹𝑦) → (∃𝑧𝑦(𝐴𝐹𝑦𝑦 = 𝑧) → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3215, 31syl5bi 243 . . . . . 6 ((𝑥𝑦𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3323, 32exlimi 2207 . . . . 5 (∃𝑦(𝑥𝑦𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧))))
3433imp 407 . . . 4 ((∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)))
3517, 34impbii 210 . . 3 (∃𝑧(𝑥𝑧 ∧ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑧)) ↔ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
361, 35bitri 276 . 2 (𝑥 ∈ (𝐹𝐴) ↔ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
3736abbi2i 2950 1 (𝐹𝐴) = {𝑥 ∣ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  ∃!weu 2646  {cab 2796   class class class wbr 5057  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by: (None)
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