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Theorem fvmptn 6567
 Description: This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class 𝐶 it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 6542. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
Hypotheses
Ref Expression
fvmptn.1 (𝑥 = 𝐷𝐵 = 𝐶)
fvmptn.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptn 𝐶 ∈ V → (𝐹𝐷) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptn
StepHypRef Expression
1 nfcv 2934 . 2 𝑥𝐷
2 nfcv 2934 . 2 𝑥𝐶
3 fvmptn.1 . 2 (𝑥 = 𝐷𝐵 = 𝐶)
4 fvmptn.2 . 2 𝐹 = (𝑥𝐴𝐵)
51, 2, 3, 4fvmptnf 6565 1 𝐶 ∈ V → (𝐹𝐷) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ∅c0 4141   ↦ cmpt 4967  ‘cfv 6137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-fv 6145 This theorem is referenced by: (None)
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