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Theorem fvmptmap 8441
Description: Special case of fvmpt 6759 for operator theorems. (Contributed by NM, 27-Nov-2007.)
Hypotheses
Ref Expression
fvmptmap.1 𝐶 ∈ V
fvmptmap.2 𝐷 ∈ V
fvmptmap.3 𝑅 ∈ V
fvmptmap.4 (𝑥 = 𝐴𝐵 = 𝐶)
fvmptmap.5 𝐹 = (𝑥 ∈ (𝑅m 𝐷) ↦ 𝐵)
Assertion
Ref Expression
fvmptmap (𝐴:𝐷𝑅 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptmap
StepHypRef Expression
1 fvmptmap.3 . . 3 𝑅 ∈ V
2 fvmptmap.2 . . 3 𝐷 ∈ V
31, 2elmap 8431 . 2 (𝐴 ∈ (𝑅m 𝐷) ↔ 𝐴:𝐷𝑅)
4 fvmptmap.4 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
5 fvmptmap.5 . . 3 𝐹 = (𝑥 ∈ (𝑅m 𝐷) ↦ 𝐵)
6 fvmptmap.1 . . 3 𝐶 ∈ V
74, 5, 6fvmpt 6759 . 2 (𝐴 ∈ (𝑅m 𝐷) → (𝐹𝐴) = 𝐶)
83, 7sylbir 238 1 (𝐴:𝐷𝑅 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  cmpt 5132  wf 6339  cfv 6343  (class class class)co 7149  m cmap 8402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404
This theorem is referenced by:  itg2val  24335  nmopval  29642  nmfnval  29662  eigvecval  29682  eigvalfval  29683  specval  29684
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