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Theorem fvmptnf 6288
Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 6289 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvmptf.1 𝑥𝐴
fvmptf.2 𝑥𝐶
fvmptf.3 (𝑥 = 𝐴𝐵 = 𝐶)
fvmptf.4 𝐹 = (𝑥𝐷𝐵)
Assertion
Ref Expression
fvmptnf 𝐶 ∈ V → (𝐹𝐴) = ∅)
Distinct variable group:   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptnf
StepHypRef Expression
1 fvmptf.4 . . . . 5 𝐹 = (𝑥𝐷𝐵)
21dmmptss 5619 . . . 4 dom 𝐹𝐷
32sseli 3591 . . 3 (𝐴 ∈ dom 𝐹𝐴𝐷)
4 eqid 2620 . . . . . . 7 (𝑥𝐷 ↦ ( I ‘𝐵)) = (𝑥𝐷 ↦ ( I ‘𝐵))
51, 4fvmptex 6281 . . . . . 6 (𝐹𝐴) = ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴)
6 fvex 6188 . . . . . . 7 ( I ‘𝐶) ∈ V
7 fvmptf.1 . . . . . . . 8 𝑥𝐴
8 nfcv 2762 . . . . . . . . 9 𝑥 I
9 fvmptf.2 . . . . . . . . 9 𝑥𝐶
108, 9nffv 6185 . . . . . . . 8 𝑥( I ‘𝐶)
11 fvmptf.3 . . . . . . . . 9 (𝑥 = 𝐴𝐵 = 𝐶)
1211fveq2d 6182 . . . . . . . 8 (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
137, 10, 12, 4fvmptf 6287 . . . . . . 7 ((𝐴𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
146, 13mpan2 706 . . . . . 6 (𝐴𝐷 → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
155, 14syl5eq 2666 . . . . 5 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
16 fvprc 6172 . . . . 5 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1715, 16sylan9eq 2674 . . . 4 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
1817expcom 451 . . 3 𝐶 ∈ V → (𝐴𝐷 → (𝐹𝐴) = ∅))
193, 18syl5 34 . 2 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅))
20 ndmfv 6205 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2119, 20pm2.61d1 171 1 𝐶 ∈ V → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  wnfc 2749  Vcvv 3195  c0 3907  cmpt 4720   I cid 5013  dom cdm 5104  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884
This theorem is referenced by:  fvmptn  6289  rdgsucmptnf  7510  frsucmptn  7519
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