Theorem fvmptss2 5496

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 5435	. 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)
2		fvmptss2.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funmpt2 5162	. . . . 5 ⊢ Fun 𝐹
4		funrel 5140	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Rel 𝐹
6	5	brrelex1i 4582	. . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V)
7		nfcv 2281	. . . 4 ⊢ Ⅎ𝑥𝐷
8		nfmpt1 4021	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	nfcxfr 2278	. . . . . 6 ⊢ Ⅎ𝑥𝐹
10		nfcv 2281	. . . . . 6 ⊢ Ⅎ𝑥𝑦
11	7, 9, 10	nfbr 3974	. . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦
12		nfv 1508	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶
13	11, 12	nfim 1551	. . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
14		breq1 3932	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦))
15		fvmptss2.1	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
16	15	sseq2d 3127	. . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶))
17	14, 16	imbi12d 233	. . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)))
18		df-br 3930	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19		opabid 4179	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
20		eqimss 3151	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 ⊆ 𝐵)
21	20	adantl 275	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ⊆ 𝐵)
22	19, 21	sylbi 120	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} → 𝑦 ⊆ 𝐵)
23		df-mpt 3991	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
24	2, 23	eqtri 2160	. . . . . 6 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
25	22, 24	eleq2s 2234	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ⊆ 𝐵)
26	18, 25	sylbi 120	. . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵)
27	7, 13, 17, 26	vtoclgf 2744	. . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))
28	6, 27	mpcom 36	. 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)
29	1, 28	mpg 1427	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929  {copab 3988   ↦ cmpt 3989  Rel wrel 4544  Fun wfun 5117  ‘cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-iota 5088  df-fun 5125  df-fv 5131

This theorem is referenced by:  mptfvex  5506