Theorem fvmptss2 5702

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 5637	. 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)
2		fvmptss2.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funmpt2 5353	. . . . 5 ⊢ Fun 𝐹
4		funrel 5331	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Rel 𝐹
6	5	brrelex1i 4759	. . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V)
7		nfcv 2372	. . . 4 ⊢ Ⅎ𝑥𝐷
8		nfmpt1 4176	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	nfcxfr 2369	. . . . . 6 ⊢ Ⅎ𝑥𝐹
10		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑥𝑦
11	7, 9, 10	nfbr 4129	. . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦
12		nfv 1574	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶
13	11, 12	nfim 1618	. . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
14		breq1 4085	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦))
15		fvmptss2.1	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
16	15	sseq2d 3254	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶))
17	14, 16	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)))
18		df-br 4083	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19		opabid 4343	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
20		eqimss 3278	. . . . . . . 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 4146	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
24	2, 23	eqtri 2250	. . . . . 6 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
25	22, 24	eleq2s 2324	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ⊆ 𝐵)
26	18, 25	sylbi 121	. . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵)
27	7, 13, 17, 26	vtoclgf 2859	. . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))
28	6, 27	mpcom 36	. 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
29	1, 28	mpg 1497	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4082  {copab 4143   ↦ cmpt 4144  Rel wrel 4721  Fun wfun 5308  ‘cfv 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-iota 5274  df-fun 5316  df-fv 5322

This theorem is referenced by:  mptfvex  5713