Theorem fvmptss2 5591

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 5529	. 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)
2		fvmptss2.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funmpt2 5255	. . . . 5 ⊢ Fun 𝐹
4		funrel 5233	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Rel 𝐹
6	5	brrelex1i 4669	. . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V)
7		nfcv 2319	. . . 4 ⊢ Ⅎ𝑥𝐷
8		nfmpt1 4096	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	nfcxfr 2316	. . . . . 6 ⊢ Ⅎ𝑥𝐹
10		nfcv 2319	. . . . . 6 ⊢ Ⅎ𝑥𝑦
11	7, 9, 10	nfbr 4049	. . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦
12		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶
13	11, 12	nfim 1572	. . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
14		breq1 4006	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦))
15		fvmptss2.1	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
16	15	sseq2d 3185	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶))
17	14, 16	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)))
18		df-br 4004	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19		opabid 4257	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
20		eqimss 3209	. . . . . . . 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 4066	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
24	2, 23	eqtri 2198	. . . . . 6 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
25	22, 24	eleq2s 2272	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ⊆ 𝐵)
26	18, 25	sylbi 121	. . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵)
27	7, 13, 17, 26	vtoclgf 2795	. . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))
28	6, 27	mpcom 36	. 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
29	1, 28	mpg 1451	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2737   ⊆ wss 3129  ⟨cop 3595   class class class wbr 4003  {copab 4063   ↦ cmpt 4064  Rel wrel 4631  Fun wfun 5210  ‘cfv 5216

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-iota 5178  df-fun 5218  df-fv 5224

This theorem is referenced by:  mptfvex  5601