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Theorem fvmptt 5205
Description: Closed theorem form of fvmpt 5192. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
fvmptt ((x(x = AB = 𝐶) 𝐹 = (x 𝐷B) (A 𝐷 𝐶 𝑉)) → (𝐹A) = 𝐶)
Distinct variable groups:   x,A   x,𝐶   x,𝐷
Allowed substitution hints:   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmptt
StepHypRef Expression
1 simp2 904 . . 3 ((x(x = AB = 𝐶) 𝐹 = (x 𝐷B) (A 𝐷 𝐶 𝑉)) → 𝐹 = (x 𝐷B))
21fveq1d 5123 . 2 ((x(x = AB = 𝐶) 𝐹 = (x 𝐷B) (A 𝐷 𝐶 𝑉)) → (𝐹A) = ((x 𝐷B)‘A))
3 risset 2346 . . . . 5 (A 𝐷x 𝐷 x = A)
4 elex 2560 . . . . . 6 (𝐶 𝑉𝐶 V)
5 nfa1 1431 . . . . . . 7 xx(x = AB = 𝐶)
6 nfv 1418 . . . . . . . 8 x 𝐶 V
7 nffvmpt1 5129 . . . . . . . . 9 x((x 𝐷B)‘A)
87nfeq1 2184 . . . . . . . 8 x((x 𝐷B)‘A) = 𝐶
96, 8nfim 1461 . . . . . . 7 x(𝐶 V → ((x 𝐷B)‘A) = 𝐶)
10 simprl 483 . . . . . . . . . . . . 13 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → x 𝐷)
11 simplr 482 . . . . . . . . . . . . . 14 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → B = 𝐶)
12 simprr 484 . . . . . . . . . . . . . 14 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → 𝐶 V)
1311, 12eqeltrd 2111 . . . . . . . . . . . . 13 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → B V)
14 eqid 2037 . . . . . . . . . . . . . 14 (x 𝐷B) = (x 𝐷B)
1514fvmpt2 5197 . . . . . . . . . . . . 13 ((x 𝐷 B V) → ((x 𝐷B)‘x) = B)
1610, 13, 15syl2anc 391 . . . . . . . . . . . 12 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → ((x 𝐷B)‘x) = B)
17 simpll 481 . . . . . . . . . . . . 13 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → x = A)
1817fveq2d 5125 . . . . . . . . . . . 12 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → ((x 𝐷B)‘x) = ((x 𝐷B)‘A))
1916, 18, 113eqtr3d 2077 . . . . . . . . . . 11 (((x = A B = 𝐶) (x 𝐷 𝐶 V)) → ((x 𝐷B)‘A) = 𝐶)
2019exp43 354 . . . . . . . . . 10 (x = A → (B = 𝐶 → (x 𝐷 → (𝐶 V → ((x 𝐷B)‘A) = 𝐶))))
2120a2i 11 . . . . . . . . 9 ((x = AB = 𝐶) → (x = A → (x 𝐷 → (𝐶 V → ((x 𝐷B)‘A) = 𝐶))))
2221com23 72 . . . . . . . 8 ((x = AB = 𝐶) → (x 𝐷 → (x = A → (𝐶 V → ((x 𝐷B)‘A) = 𝐶))))
2322sps 1427 . . . . . . 7 (x(x = AB = 𝐶) → (x 𝐷 → (x = A → (𝐶 V → ((x 𝐷B)‘A) = 𝐶))))
245, 9, 23rexlimd 2424 . . . . . 6 (x(x = AB = 𝐶) → (x 𝐷 x = A → (𝐶 V → ((x 𝐷B)‘A) = 𝐶)))
254, 24syl7 63 . . . . 5 (x(x = AB = 𝐶) → (x 𝐷 x = A → (𝐶 𝑉 → ((x 𝐷B)‘A) = 𝐶)))
263, 25syl5bi 141 . . . 4 (x(x = AB = 𝐶) → (A 𝐷 → (𝐶 𝑉 → ((x 𝐷B)‘A) = 𝐶)))
2726imp32 244 . . 3 ((x(x = AB = 𝐶) (A 𝐷 𝐶 𝑉)) → ((x 𝐷B)‘A) = 𝐶)
28273adant2 922 . 2 ((x(x = AB = 𝐶) 𝐹 = (x 𝐷B) (A 𝐷 𝐶 𝑉)) → ((x 𝐷B)‘A) = 𝐶)
292, 28eqtrd 2069 1 ((x(x = AB = 𝐶) 𝐹 = (x 𝐷B) (A 𝐷 𝐶 𝑉)) → (𝐹A) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884  wal 1240   = wceq 1242   wcel 1390  wrex 2301  Vcvv 2551  cmpt 3809  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by: (None)
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