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Theorem fvmptt 5289
Description: Closed theorem form of fvmpt 5276. (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 916 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → 𝐹 = (𝑥𝐷𝐵))
21fveq1d 5207 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
3 risset 2369 . . . . 5 (𝐴𝐷 ↔ ∃𝑥𝐷 𝑥 = 𝐴)
4 elex 2583 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
5 nfa1 1450 . . . . . . 7 𝑥𝑥(𝑥 = 𝐴𝐵 = 𝐶)
6 nfv 1437 . . . . . . . 8 𝑥 𝐶 ∈ V
7 nffvmpt1 5213 . . . . . . . . 9 𝑥((𝑥𝐷𝐵)‘𝐴)
87nfeq1 2203 . . . . . . . 8 𝑥((𝑥𝐷𝐵)‘𝐴) = 𝐶
96, 8nfim 1480 . . . . . . 7 𝑥(𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
10 simprl 491 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥𝐷)
11 simplr 490 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 = 𝐶)
12 simprr 492 . . . . . . . . . . . . . 14 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐶 ∈ V)
1311, 12eqeltrd 2130 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝐵 ∈ V)
14 eqid 2056 . . . . . . . . . . . . . 14 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1514fvmpt2 5281 . . . . . . . . . . . . 13 ((𝑥𝐷𝐵 ∈ V) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1610, 13, 15syl2anc 397 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
17 simpll 489 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → 𝑥 = 𝐴)
1817fveq2d 5209 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
1916, 18, 113eqtr3d 2096 . . . . . . . . . . 11 (((𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝑥𝐷𝐶 ∈ V)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
2019exp43 358 . . . . . . . . . 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 76 . . . . . . . 8 ((𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
2322sps 1446 . . . . . . 7 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝑥𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶))))
245, 9, 23rexlimd 2447 . . . . . 6 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (∃𝑥𝐷 𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
254, 24syl7 67 . . . . 5 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (∃𝑥𝐷 𝑥 = 𝐴 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
263, 25syl5bi 145 . . . 4 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → (𝐴𝐷 → (𝐶𝑉 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)))
2726imp32 248 . . 3 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
28273adant2 934 . 2 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
292, 28eqtrd 2088 1 ((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896  wal 1257   = wceq 1259  wcel 1409  wrex 2324  Vcvv 2574  cmpt 3845  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937
This theorem is referenced by: (None)
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