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Theorem fvmptmap 7838
Description: Special case of fvmpt 6239 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 7830 . 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 6239 . 2 (𝐴 ∈ (𝑅𝑚 𝐷) → (𝐹𝐴) = 𝐶)
83, 7sylbir 225 1 (𝐴:𝐷𝑅 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804
This theorem is referenced by:  itg2val  23401  nmopval  28561  nmfnval  28581  eigvecval  28601  eigvalfval  28602  specval  28603
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