Theorem fvmptnf 6951

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 6954 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 6188	. . . 4 ⊢ dom 𝐹 ⊆ 𝐷
3	2	sseli 3925	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷)
4		eqid 2731	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))
5	1, 4	fvmptex 6943	. . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴)
6		fvex 6835	. . . . . . 7 ⊢ ( I ‘𝐶) ∈ V
7		fvmptf.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
8		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑥 I
9		fvmptf.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
10	8, 9	nffv 6832	. . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶)
11		fvmptf.3	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	fveq2d 6826	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
13	7, 10, 12, 4	fvmptf 6950	. . . . . . 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 2778	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))
16		fvprc 6814	. . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅)
17	15, 16	sylan9eq 2786	. . . 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 6854	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
21	19, 20	pm2.61d1 180	1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879  Vcvv 3436  ∅c0 4280   ↦ cmpt 5170   I cid 5508  dom cdm 5614  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  fvmptn  6954  rdgsucmptnf  8348  frsucmptn  8358