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Theorem fvmptnf 6781
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 6783 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 6082 . . . 4 dom 𝐹𝐷
32sseli 3949 . . 3 (𝐴 ∈ dom 𝐹𝐴𝐷)
4 eqid 2824 . . . . . . 7 (𝑥𝐷 ↦ ( I ‘𝐵)) = (𝑥𝐷 ↦ ( I ‘𝐵))
51, 4fvmptex 6773 . . . . . 6 (𝐹𝐴) = ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴)
6 fvex 6674 . . . . . . 7 ( I ‘𝐶) ∈ V
7 fvmptf.1 . . . . . . . 8 𝑥𝐴
8 nfcv 2982 . . . . . . . . 9 𝑥 I
9 fvmptf.2 . . . . . . . . 9 𝑥𝐶
108, 9nffv 6671 . . . . . . . 8 𝑥( I ‘𝐶)
11 fvmptf.3 . . . . . . . . 9 (𝑥 = 𝐴𝐵 = 𝐶)
1211fveq2d 6665 . . . . . . . 8 (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
137, 10, 12, 4fvmptf 6780 . . . . . . 7 ((𝐴𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
146, 13mpan2 690 . . . . . 6 (𝐴𝐷 → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
155, 14syl5eq 2871 . . . . 5 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
16 fvprc 6654 . . . . 5 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1715, 16sylan9eq 2879 . . . 4 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
1817expcom 417 . . 3 𝐶 ∈ V → (𝐴𝐷 → (𝐹𝐴) = ∅))
193, 18syl5 34 . 2 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅))
20 ndmfv 6691 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2119, 20pm2.61d1 183 1 𝐶 ∈ V → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  wnfc 2962  Vcvv 3480  c0 4276  cmpt 5132   I cid 5446  dom cdm 5542  cfv 6343
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
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-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  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-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351
This theorem is referenced by:  fvmptn  6783  rdgsucmptnf  8061  frsucmptn  8070
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