Theorem fvmptnf 6990

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 6993 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 6214	. . . 4 ⊢ dom 𝐹 ⊆ 𝐷
3	2	sseli 3942	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷)
4		eqid 2729	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))
5	1, 4	fvmptex 6982	. . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴)
6		fvex 6871	. . . . . . 7 ⊢ ( I ‘𝐶) ∈ V
7		fvmptf.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
8		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥 I
9		fvmptf.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
10	8, 9	nffv 6868	. . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶)
11		fvmptf.3	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	fveq2d 6862	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
13	7, 10, 12, 4	fvmptf 6989	. . . . . . 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 2776	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))
16		fvprc 6850	. . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅)
17	15, 16	sylan9eq 2784	. . . 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 6893	. 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 2109  Ⅎwnfc 2876  Vcvv 3447  ∅c0 4296   ↦ cmpt 5188   I cid 5532  dom cdm 5638  ‘cfv 6511

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  fvmptn  6993  rdgsucmptnf  8397  frsucmptn  8407