Theorem fvmptss2 5730

Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)

Hypotheses

Ref	Expression
fvmptss2.1	⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
fvmptss2.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fvmptss2	⊢ (𝐹‘𝐷) ⊆ 𝐶

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

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

Proof of Theorem fvmptss2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvss 5662	. 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)
2		fvmptss2.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funmpt2 5372	. . . . 5 ⊢ Fun 𝐹
4		funrel 5350	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Rel 𝐹
6	5	brrelex1i 4775	. . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V)
7		nfcv 2375	. . . 4 ⊢ Ⅎ𝑥𝐷
8		nfmpt1 4187	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	nfcxfr 2372	. . . . . 6 ⊢ Ⅎ𝑥𝐹
10		nfcv 2375	. . . . . 6 ⊢ Ⅎ𝑥𝑦
11	7, 9, 10	nfbr 4140	. . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦
12		nfv 1577	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶
13	11, 12	nfim 1621	. . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
14		breq1 4096	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦))
15		fvmptss2.1	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
16	15	sseq2d 3258	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶))
17	14, 16	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)))
18		df-br 4094	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19		opabid 4356	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
20		eqimss 3282	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 ⊆ 𝐵)
21	20	adantl 277	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ⊆ 𝐵)
22	19, 21	sylbi 121	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} → 𝑦 ⊆ 𝐵)
23		df-mpt 4157	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
24	2, 23	eqtri 2252	. . . . . 6 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
25	22, 24	eleq2s 2326	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ⊆ 𝐵)
26	18, 25	sylbi 121	. . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵)
27	7, 13, 17, 26	vtoclgf 2863	. . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))
28	6, 27	mpcom 36	. 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
29	1, 28	mpg 1500	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ⊆ wss 3201  ⟨cop 3676   class class class wbr 4093  {copab 4154   ↦ cmpt 4155  Rel wrel 4736  Fun wfun 5327  ‘cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-iota 5293  df-fun 5335  df-fv 5341

This theorem is referenced by:  mptfvex  5741