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Theorem fvmptmap 8159
Description: Special case of fvmpt 6529 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 8151 . 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 6529 . 2 (𝐴 ∈ (𝑅𝑚 𝐷) → (𝐹𝐴) = 𝐶)
83, 7sylbir 227 1 (𝐴:𝐷𝑅 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  Vcvv 3414  cmpt 4952  wf 6119  cfv 6123  (class class class)co 6905  𝑚 cmap 8122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124
This theorem is referenced by:  itg2val  23894  nmopval  29270  nmfnval  29290  eigvecval  29310  eigvalfval  29311  specval  29312
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