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Theorem fvmptnf 7038
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 7041 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 6261 . . . 4 dom 𝐹𝐷
32sseli 3979 . . 3 (𝐴 ∈ dom 𝐹𝐴𝐷)
4 eqid 2737 . . . . . . 7 (𝑥𝐷 ↦ ( I ‘𝐵)) = (𝑥𝐷 ↦ ( I ‘𝐵))
51, 4fvmptex 7030 . . . . . 6 (𝐹𝐴) = ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴)
6 fvex 6919 . . . . . . 7 ( I ‘𝐶) ∈ V
7 fvmptf.1 . . . . . . . 8 𝑥𝐴
8 nfcv 2905 . . . . . . . . 9 𝑥 I
9 fvmptf.2 . . . . . . . . 9 𝑥𝐶
108, 9nffv 6916 . . . . . . . 8 𝑥( I ‘𝐶)
11 fvmptf.3 . . . . . . . . 9 (𝑥 = 𝐴𝐵 = 𝐶)
1211fveq2d 6910 . . . . . . . 8 (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
137, 10, 12, 4fvmptf 7037 . . . . . . 7 ((𝐴𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
146, 13mpan2 691 . . . . . 6 (𝐴𝐷 → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
155, 14eqtrid 2789 . . . . 5 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
16 fvprc 6898 . . . . 5 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1715, 16sylan9eq 2797 . . . 4 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
1817expcom 413 . . 3 𝐶 ∈ V → (𝐴𝐷 → (𝐹𝐴) = ∅))
193, 18syl5 34 . 2 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅))
20 ndmfv 6941 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2119, 20pm2.61d1 180 1 𝐶 ∈ V → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  wnfc 2890  Vcvv 3480  c0 4333  cmpt 5225   I cid 5577  dom cdm 5685  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  fvmptn  7041  rdgsucmptnf  8469  frsucmptn  8479
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