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Theorem fvmptt 6780
Description: Closed theorem form of fvmpt 6761. (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 1129 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → 𝐹 = (𝑥𝐷𝐵))
21fveq1d 6665 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
3 risset 3264 . . . . 5 (𝐴𝐷 ↔ ∃𝑥𝐷 𝑥 = 𝐴)
4 elex 3510 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
5 nfa1 2146 . . . . . . 7 𝑥𝑥(𝑥 = 𝐴𝐵 = 𝐶)
6 nfv 1906 . . . . . . . 8 𝑥 𝐶 ∈ V
7 nffvmpt1 6674 . . . . . . . . 9 𝑥((𝑥𝐷𝐵)‘𝐴)
87nfeq1 2990 . . . . . . . 8 𝑥((𝑥𝐷𝐵)‘𝐴) = 𝐶
96, 8nfim 1888 . . . . . . 7 𝑥(𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
10 simprl 767 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥𝐷)
11 simplr 765 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 = 𝐶)
12 simprr 769 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐶 ∈ V)
1311, 12eqeltrd 2910 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 ∈ V)
14 eqid 2818 . . . . . . . . . . . . . 14 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1514fvmpt2 6771 . . . . . . . . . . . . 13 ((𝑥𝐷𝐵 ∈ V) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1610, 13, 15syl2anc 584 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
17 simpll 763 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥 = 𝐴)
1817fveq2d 6667 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
1916, 18, 113eqtr3d 2861 . . . . . . . . . . 11 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
2019exp43 437 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐵 = 𝐶 → (𝑥𝐷 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
2120a2i 14 . . . . . . . . 9 ((𝑥 = 𝐴𝐵 = 𝐶) → (𝑥 = 𝐴 → (𝑥𝐷 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
2221com23 86 . . . . . . . 8 ((𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
2322sps 2174 . . . . . . 7 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
245, 9, 23rexlimd 3314 . . . . . 6 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (∃𝑥𝐷 𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
254, 24syl7 74 . . . . 5 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (∃𝑥𝐷 𝑥 = 𝐴 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
263, 25syl5bi 243 . . . 4 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝐴𝐷 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
2726imp32 419 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
28273adant2 1123 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
292, 28eqtrd 2853 1 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wal 1526   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  cmpt 5137  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  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  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-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by: (None)
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