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Theorem fvmptt 6561
Description: Closed theorem form of fvmpt 6542. (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 1128 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → 𝐹 = (𝑥𝐷𝐵))
21fveq1d 6448 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
3 risset 3247 . . . . 5 (𝐴𝐷 ↔ ∃𝑥𝐷 𝑥 = 𝐴)
4 elex 3414 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
5 nfa1 2145 . . . . . . 7 𝑥𝑥(𝑥 = 𝐴𝐵 = 𝐶)
6 nfv 1957 . . . . . . . 8 𝑥 𝐶 ∈ V
7 nffvmpt1 6457 . . . . . . . . 9 𝑥((𝑥𝐷𝐵)‘𝐴)
87nfeq1 2947 . . . . . . . 8 𝑥((𝑥𝐷𝐵)‘𝐴) = 𝐶
96, 8nfim 1943 . . . . . . 7 𝑥(𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
10 simprl 761 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥𝐷)
11 simplr 759 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 = 𝐶)
12 simprr 763 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐶 ∈ V)
1311, 12eqeltrd 2859 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 ∈ V)
14 eqid 2778 . . . . . . . . . . . . . 14 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1514fvmpt2 6552 . . . . . . . . . . . . 13 ((𝑥𝐷𝐵 ∈ V) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1610, 13, 15syl2anc 579 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
17 simpll 757 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥 = 𝐴)
1817fveq2d 6450 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
1916, 18, 113eqtr3d 2822 . . . . . . . . . . 11 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
2019exp43 429 . . . . . . . . . 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 2169 . . . . . . 7 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
245, 9, 23rexlimd 3208 . . . . . 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 234 . . . 4 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝐴𝐷 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
2726imp32 411 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
28273adant2 1122 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
292, 28eqtrd 2814 1 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071  wal 1599   = wceq 1601  wcel 2107  wrex 3091  Vcvv 3398  cmpt 4965  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143
This theorem is referenced by: (None)
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