Theorem fvmptss2 5721

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 5653	. 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)
2		fvmptss2.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funmpt2 5365	. . . . 5 ⊢ Fun 𝐹
4		funrel 5343	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Rel 𝐹
6	5	brrelex1i 4769	. . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V)
7		nfcv 2374	. . . 4 ⊢ Ⅎ𝑥𝐷
8		nfmpt1 4182	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	nfcxfr 2371	. . . . . 6 ⊢ Ⅎ𝑥𝐹
10		nfcv 2374	. . . . . 6 ⊢ Ⅎ𝑥𝑦
11	7, 9, 10	nfbr 4135	. . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦
12		nfv 1576	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶
13	11, 12	nfim 1620	. . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
14		breq1 4091	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦))
15		fvmptss2.1	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
16	15	sseq2d 3257	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶))
17	14, 16	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)))
18		df-br 4089	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19		opabid 4350	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
20		eqimss 3281	. . . . . . . 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 4152	. . . . . . 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 2862	. . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))
28	6, 27	mpcom 36	. 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
29	1, 28	mpg 1499	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088  {copab 4149   ↦ cmpt 4150  Rel wrel 4730  Fun wfun 5320  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-iota 5286  df-fun 5328  df-fv 5334

This theorem is referenced by:  mptfvex  5732