Theorem fvmptnf 7008

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 7011 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 6230	. . . 4 ⊢ dom 𝐹 ⊆ 𝐷
3	2	sseli 3954	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷)
4		eqid 2735	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))
5	1, 4	fvmptex 7000	. . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴)
6		fvex 6889	. . . . . . 7 ⊢ ( I ‘𝐶) ∈ V
7		fvmptf.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
8		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑥 I
9		fvmptf.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
10	8, 9	nffv 6886	. . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶)
11		fvmptf.3	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	fveq2d 6880	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
13	7, 10, 12, 4	fvmptf 7007	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
14	6, 13	mpan2 691	. . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
15	5, 14	eqtrid 2782	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))
16		fvprc 6868	. . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅)
17	15, 16	sylan9eq 2790	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅)
18	17	expcom 413	. . 3 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ∅))
19	3, 18	syl5 34	. 2 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅))
20		ndmfv 6911	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
21	19, 20	pm2.61d1 180	1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2883  Vcvv 3459  ∅c0 4308   ↦ cmpt 5201   I cid 5547  dom cdm 5654  ‘cfv 6531

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is referenced by:  fvmptn  7011  rdgsucmptnf  8443  frsucmptn  8453