ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvmptt GIF version

Theorem fvmptt 5628
Description: Closed theorem form of fvmpt 5614. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
fvmptt ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (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 1000 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → 𝐹 = (𝑥𝐷𝐵))
21fveq1d 5536 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
3 risset 2518 . . . . 5 (𝐴𝐷 ↔ ∃𝑥𝐷 𝑥 = 𝐴)
4 elex 2763 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
5 nfa1 1552 . . . . . . 7 𝑥𝑥(𝑥 = 𝐴𝐵 = 𝐶)
6 nfv 1539 . . . . . . . 8 𝑥 𝐶 ∈ V
7 nffvmpt1 5545 . . . . . . . . 9 𝑥((𝑥𝐷𝐵)‘𝐴)
87nfeq1 2342 . . . . . . . 8 𝑥((𝑥𝐷𝐵)‘𝐴) = 𝐶
96, 8nfim 1583 . . . . . . 7 𝑥(𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
10 simprl 529 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥𝐷)
11 simplr 528 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 = 𝐶)
12 simprr 531 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐶 ∈ V)
1311, 12eqeltrd 2266 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 ∈ V)
14 eqid 2189 . . . . . . . . . . . . . 14 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1514fvmpt2 5620 . . . . . . . . . . . . 13 ((𝑥𝐷𝐵 ∈ V) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1610, 13, 15syl2anc 411 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
17 simpll 527 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥 = 𝐴)
1817fveq2d 5538 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
1916, 18, 113eqtr3d 2230 . . . . . . . . . . 11 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
2019exp43 372 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐵 = 𝐶 → (𝑥𝐷 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
2120a2i 11 . . . . . . . . 9 ((𝑥 = 𝐴𝐵 = 𝐶) → (𝑥 = 𝐴 → (𝑥𝐷 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
2221com23 78 . . . . . . . 8 ((𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
2322sps 1548 . . . . . . 7 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
245, 9, 23rexlimd 2604 . . . . . 6 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (∃𝑥𝐷 𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
254, 24syl7 69 . . . . 5 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (∃𝑥𝐷 𝑥 = 𝐴 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
263, 25biimtrid 152 . . . 4 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝐴𝐷 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
2726imp32 257 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
28273adant2 1018 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
292, 28eqtrd 2222 1 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980  wal 1362   = wceq 1364  wcel 2160  wrex 2469  Vcvv 2752  cmpt 4079  cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator