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Theorem fvmptdv2 5203
Description: Alternate deduction version of fvmpt 5192, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdv2.1 (φA 𝐷)
fvmptdv2.2 ((φ x = A) → B 𝑉)
fvmptdv2.3 ((φ x = A) → B = 𝐶)
Assertion
Ref Expression
fvmptdv2 (φ → (𝐹 = (x 𝐷B) → (𝐹A) = 𝐶))
Distinct variable groups:   x,A   x,𝐶   x,𝐷   φ,x
Allowed substitution hints:   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmptdv2
StepHypRef Expression
1 eqidd 2038 . . 3 (φ → (x 𝐷B) = (x 𝐷B))
2 fvmptdv2.3 . . 3 ((φ x = A) → B = 𝐶)
3 fvmptdv2.1 . . 3 (φA 𝐷)
4 elex 2560 . . . . . 6 (A 𝐷A V)
53, 4syl 14 . . . . 5 (φA V)
6 isset 2555 . . . . 5 (A V ↔ x x = A)
75, 6sylib 127 . . . 4 (φx x = A)
8 fvmptdv2.2 . . . . . 6 ((φ x = A) → B 𝑉)
9 elex 2560 . . . . . 6 (B 𝑉B V)
108, 9syl 14 . . . . 5 ((φ x = A) → B V)
112, 10eqeltrrd 2112 . . . 4 ((φ x = A) → 𝐶 V)
127, 11exlimddv 1775 . . 3 (φ𝐶 V)
131, 2, 3, 12fvmptd 5196 . 2 (φ → ((x 𝐷B)‘A) = 𝐶)
14 fveq1 5120 . . 3 (𝐹 = (x 𝐷B) → (𝐹A) = ((x 𝐷B)‘A))
1514eqeq1d 2045 . 2 (𝐹 = (x 𝐷B) → ((𝐹A) = 𝐶 ↔ ((x 𝐷B)‘A) = 𝐶))
1613, 15syl5ibrcom 146 1 (φ → (𝐹 = (x 𝐷B) → (𝐹A) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cmpt 3809  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by: (None)
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