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Theorem fv3 5341
 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(x y AFy) ∃!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 5326 . . 3 (x (FA) ↔ z(x z y(AFyy = z)))
2 bi2 189 . . . . . . . . . 10 ((AFyy = z) → (y = zAFy))
32alimi 1559 . . . . . . . . 9 (y(AFyy = z) → y(y = zAFy))
4 vex 2862 . . . . . . . . . 10 z V
5 breq2 4643 . . . . . . . . . 10 (y = z → (AFyAFz))
64, 5ceqsalv 2885 . . . . . . . . 9 (y(y = zAFy) ↔ AFz)
73, 6sylib 188 . . . . . . . 8 (y(AFyy = z) → AFz)
87anim2i 552 . . . . . . 7 ((x z y(AFyy = z)) → (x z AFz))
98eximi 1576 . . . . . 6 (z(x z y(AFyy = z)) → z(x z AFz))
10 elequ2 1715 . . . . . . . 8 (z = y → (x zx y))
11 breq2 4643 . . . . . . . 8 (z = y → (AFzAFy))
1210, 11anbi12d 691 . . . . . . 7 (z = y → ((x z AFz) ↔ (x y AFy)))
1312cbvexv 2003 . . . . . 6 (z(x z AFz) ↔ y(x y AFy))
149, 13sylib 188 . . . . 5 (z(x z y(AFyy = z)) → y(x y AFy))
15 19.40 1609 . . . . . . 7 (z(x z y(AFyy = z)) → (z x z zy(AFyy = z)))
1615simprd 449 . . . . . 6 (z(x z y(AFyy = z)) → zy(AFyy = z))
17 df-eu 2208 . . . . . 6 (∃!y AFyzy(AFyy = z))
1816, 17sylibr 203 . . . . 5 (z(x z y(AFyy = z)) → ∃!y AFy)
1914, 18jca 518 . . . 4 (z(x z y(AFyy = z)) → (y(x y AFy) ∃!y AFy))
20 nfeu1 2214 . . . . . . 7 y∃!y AFy
21 nfv 1619 . . . . . . . . 9 y x z
22 nfa1 1788 . . . . . . . . 9 yy(AFyy = z)
2321, 22nfan 1824 . . . . . . . 8 y(x z y(AFyy = z))
2423nfex 1843 . . . . . . 7 yz(x z y(AFyy = z))
2520, 24nfim 1813 . . . . . 6 y(∃!y AFyz(x z y(AFyy = z)))
26 bi1 178 . . . . . . . . . . . . . 14 ((AFyy = z) → (AFyy = z))
27 ax-14 1714 . . . . . . . . . . . . . 14 (y = z → (x yx z))
2826, 27syl6 29 . . . . . . . . . . . . 13 ((AFyy = z) → (AFy → (x yx z)))
2928com23 72 . . . . . . . . . . . 12 ((AFyy = z) → (x y → (AFyx z)))
3029imp3a 420 . . . . . . . . . . 11 ((AFyy = z) → ((x y AFy) → x z))
3130sps 1754 . . . . . . . . . 10 (y(AFyy = z) → ((x y AFy) → x z))
3231anc2ri 541 . . . . . . . . 9 (y(AFyy = z) → ((x y AFy) → (x z y(AFyy = z))))
3332com12 27 . . . . . . . 8 ((x y AFy) → (y(AFyy = z) → (x z y(AFyy = z))))
3433eximdv 1622 . . . . . . 7 ((x y AFy) → (zy(AFyy = z) → z(x z y(AFyy = z))))
3517, 34syl5bi 208 . . . . . 6 ((x y AFy) → (∃!y AFyz(x z y(AFyy = z))))
3625, 35exlimi 1803 . . . . 5 (y(x y AFy) → (∃!y AFyz(x z y(AFyy = z))))
3736imp 418 . . . 4 ((y(x y AFy) ∃!y AFy) → z(x z y(AFyy = z)))
3819, 37impbii 180 . . 3 (z(x z y(AFyy = z)) ↔ (y(x y AFy) ∃!y AFy))
391, 38bitri 240 . 2 (x (FA) ↔ (y(x y AFy) ∃!y AFy))
4039abbi2i 2464 1 (FA) = {x (y(x y AFy) ∃!y AFy)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339   class class class wbr 4639   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795 This theorem is referenced by:  tz6.12-2  5346
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