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Theorem fvmptnf 7037
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 7040 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 6262 . . . 4 dom 𝐹𝐷
32sseli 3990 . . 3 (𝐴 ∈ dom 𝐹𝐴𝐷)
4 eqid 2734 . . . . . . 7 (𝑥𝐷 ↦ ( I ‘𝐵)) = (𝑥𝐷 ↦ ( I ‘𝐵))
51, 4fvmptex 7029 . . . . . 6 (𝐹𝐴) = ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴)
6 fvex 6919 . . . . . . 7 ( I ‘𝐶) ∈ V
7 fvmptf.1 . . . . . . . 8 𝑥𝐴
8 nfcv 2902 . . . . . . . . 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 7036 . . . . . . 7 ((𝐴𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
146, 13mpan2 691 . . . . . 6 (𝐴𝐷 → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
155, 14eqtrid 2786 . . . . 5 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
16 fvprc 6898 . . . . 5 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1715, 16sylan9eq 2794 . . . 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 1536  wcel 2105  wnfc 2887  Vcvv 3477  c0 4338  cmpt 5230   I cid 5581  dom cdm 5688  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  fvmptn  7040  rdgsucmptnf  8467  frsucmptn  8477
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