Theorem fvmptnf 5724

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 5725 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	⊢ ℲxA
fvmptf.2	⊢ ℲxC
fvmptf.3	⊢ (x = A → B = C)
fvmptf.4	⊢ F = (x ∈ D ↦ B)

Assertion

Ref	Expression
fvmptnf	⊢ (¬ C ∈ V → (F ‘A) = ∅)

Distinct variable group:   x,D

Allowed substitution hints:   A(x)   B(x)   C(x)   F(x)

Proof of Theorem fvmptnf

Step	Hyp	Ref	Expression
1		fvmptf.4	. . . . 5 ⊢ F = (x ∈ D ↦ B)
2	1	dmmptss 5686	. . . 4 ⊢ dom F ⊆ D
3	2	sseli 3270	. . 3 ⊢ (A ∈ dom F → A ∈ D)
4		eqid 2353	. . . . . . 7 ⊢ (x ∈ D ↦ ( I ‘B)) = (x ∈ D ↦ ( I ‘B))
5	1, 4	fvmptex 5722	. . . . . 6 ⊢ (F ‘A) = ((x ∈ D ↦ ( I ‘B)) ‘A)
6		fvex 5340	. . . . . . 7 ⊢ ( I ‘C) ∈ V
7		fvmptf.1	. . . . . . . 8 ⊢ ℲxA
8		nfcv 2490	. . . . . . . . 9 ⊢ Ⅎx I
9		fvmptf.2	. . . . . . . . 9 ⊢ ℲxC
10	8, 9	nffv 5335	. . . . . . . 8 ⊢ Ⅎx( I ‘C)
11		fvmptf.3	. . . . . . . . 9 ⊢ (x = A → B = C)
12	11	fveq2d 5333	. . . . . . . 8 ⊢ (x = A → ( I ‘B) = ( I ‘C))
13	7, 10, 12, 4	fvmptf 5723	. . . . . . 7 ⊢ ((A ∈ D ∧ ( I ‘C) ∈ V) → ((x ∈ D ↦ ( I ‘B)) ‘A) = ( I ‘C))
14	6, 13	mpan2 652	. . . . . 6 ⊢ (A ∈ D → ((x ∈ D ↦ ( I ‘B)) ‘A) = ( I ‘C))
15	5, 14	syl5eq 2397	. . . . 5 ⊢ (A ∈ D → (F ‘A) = ( I ‘C))
16		fvprc 5326	. . . . 5 ⊢ (¬ C ∈ V → ( I ‘C) = ∅)
17	15, 16	sylan9eq 2405	. . . 4 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → (F ‘A) = ∅)
18	17	expcom 424	. . 3 ⊢ (¬ C ∈ V → (A ∈ D → (F ‘A) = ∅))
19	3, 18	syl5 28	. 2 ⊢ (¬ C ∈ V → (A ∈ dom F → (F ‘A) = ∅))
20		ndmfv 5350	. 2 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅)
21	19, 20	pm2.61d1 151	1 ⊢ (¬ C ∈ V → (F ‘A) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  Vcvv 2860  ∅c0 3551   I cid 4764  dom cdm 4773   ‘cfv 4782   ↦ cmpt 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796  df-mpt 5653

This theorem is referenced by:  fvmptn  5725