Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptnf Structured version   Visualization version   GIF version

Theorem fvmptnf 6554
 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 6556 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 5876 . . . 4 dom 𝐹𝐷
32sseli 3823 . . 3 (𝐴 ∈ dom 𝐹𝐴𝐷)
4 eqid 2825 . . . . . . 7 (𝑥𝐷 ↦ ( I ‘𝐵)) = (𝑥𝐷 ↦ ( I ‘𝐵))
51, 4fvmptex 6546 . . . . . 6 (𝐹𝐴) = ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴)
6 fvex 6450 . . . . . . 7 ( I ‘𝐶) ∈ V
7 fvmptf.1 . . . . . . . 8 𝑥𝐴
8 nfcv 2969 . . . . . . . . 9 𝑥 I
9 fvmptf.2 . . . . . . . . 9 𝑥𝐶
108, 9nffv 6447 . . . . . . . 8 𝑥( I ‘𝐶)
11 fvmptf.3 . . . . . . . . 9 (𝑥 = 𝐴𝐵 = 𝐶)
1211fveq2d 6441 . . . . . . . 8 (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
137, 10, 12, 4fvmptf 6553 . . . . . . 7 ((𝐴𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
146, 13mpan2 682 . . . . . 6 (𝐴𝐷 → ((𝑥𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
155, 14syl5eq 2873 . . . . 5 (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
16 fvprc 6430 . . . . 5 𝐶 ∈ V → ( I ‘𝐶) = ∅)
1715, 16sylan9eq 2881 . . . 4 ((𝐴𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹𝐴) = ∅)
1817expcom 404 . . 3 𝐶 ∈ V → (𝐴𝐷 → (𝐹𝐴) = ∅))
193, 18syl5 34 . 2 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅))
20 ndmfv 6467 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
2119, 20pm2.61d1 173 1 𝐶 ∈ V → (𝐹𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1656   ∈ wcel 2164  Ⅎwnfc 2956  Vcvv 3414  ∅c0 4146   ↦ cmpt 4954   I cid 5251  dom cdm 5346  ‘cfv 6127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-fv 6135 This theorem is referenced by:  fvmptn  6556  rdgsucmptnf  7796  frsucmptn  7805
 Copyright terms: Public domain W3C validator