Theorem fvmptnf 7010

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 7013 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 6237	. . . 4 ⊢ dom 𝐹 ⊆ 𝐷
3	2	sseli 3927	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ 𝐷)
4		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵)) = (𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))
5	1, 4	fvmptex 7002	. . . . . 6 ⊢ (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴)
6		fvex 6892	. . . . . . 7 ⊢ ( I ‘𝐶) ∈ V
7		fvmptf.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
8		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑥 I
9		fvmptf.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
10	8, 9	nffv 6889	. . . . . . . 8 ⊢ Ⅎ𝑥( I ‘𝐶)
11		fvmptf.3	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	fveq2d 6883	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ( I ‘𝐵) = ( I ‘𝐶))
13	7, 10, 12, 4	fvmptf 7009	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ ( I ‘𝐶) ∈ V) → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
14	6, 13	mpan2 704	. . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ ( I ‘𝐵))‘𝐴) = ( I ‘𝐶))
15	5, 14	eqtrid 2807	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))
16		fvprc 6871	. . . . 5 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅)
17	15, 16	sylan9eq 2815	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅)
18	17	expcom 419	. . 3 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ∅))
19	3, 18	syl5 35	. 2 ⊢ (¬ 𝐶 ∈ V → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅))
20		ndmfv 6911	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
21	19, 20	pm2.61d1 182	1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2907  Vcvv 3450  ∅c0 4279   ↦ cmpt 5186   I cid 5549  dom cdm 5655  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is used by:  fvmptn  7013  rdgsucmptnf  8419  frsucmptn  8429