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Theorem fvmptt 7011
Description: Closed theorem form of fvmpt 6991. (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 1134 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → 𝐹 = (𝑥𝐷𝐵))
21fveq1d 6886 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
3 risset 3224 . . . . 5 (𝐴𝐷 ↔ ∃𝑥𝐷 𝑥 = 𝐴)
4 elex 3487 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
5 nfa1 2140 . . . . . . 7 𝑥𝑥(𝑥 = 𝐴𝐵 = 𝐶)
6 nfv 1909 . . . . . . . 8 𝑥 𝐶 ∈ V
7 nffvmpt1 6895 . . . . . . . . 9 𝑥((𝑥𝐷𝐵)‘𝐴)
87nfeq1 2912 . . . . . . . 8 𝑥((𝑥𝐷𝐵)‘𝐴) = 𝐶
96, 8nfim 1891 . . . . . . 7 𝑥(𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
10 simprl 768 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥𝐷)
11 simplr 766 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 = 𝐶)
12 simprr 770 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐶 ∈ V)
1311, 12eqeltrd 2827 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 ∈ V)
14 eqid 2726 . . . . . . . . . . . . . 14 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1514fvmpt2 7002 . . . . . . . . . . . . 13 ((𝑥𝐷𝐵 ∈ V) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1610, 13, 15syl2anc 583 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
17 simpll 764 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥 = 𝐴)
1817fveq2d 6888 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
1916, 18, 113eqtr3d 2774 . . . . . . . . . . 11 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
2019exp43 436 . . . . . . . . . 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 2170 . . . . . . 7 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
245, 9, 23rexlimd 3257 . . . . . 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, 25biimtrid 241 . . . 4 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝐴𝐷 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
2726imp32 418 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
28273adant2 1128 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
292, 28eqtrd 2766 1 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468  cmpt 5224  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544
This theorem is referenced by: (None)
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