Theorem mptfvex 5728

Description: Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Hypothesis

Ref	Expression
fvmpt2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
mptfvex	⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘𝐶) ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

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

Proof of Theorem mptfvex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3128	. . 3 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
2		fvmpt2.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3158	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3134	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 4182	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2250	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmptss2 5717	. 2 ⊢ (𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵
9		elex 2812	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
10	9	alimi 1501	. . . . 5 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑥 𝐵 ∈ V)
11	3	nfel1 2383	. . . . . 6 ⊢ Ⅎ𝑦 𝐵 ∈ V
12	4	nfel1 2383	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ V
13	5	eleq1d 2298	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ V))
14	11, 12, 13	cbval 1800	. . . . 5 ⊢ (∀𝑥 𝐵 ∈ V ↔ ∀𝑦⦋𝑦 / 𝑥⦌𝐵 ∈ V)
15	10, 14	sylib 122	. . . 4 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑦⦋𝑦 / 𝑥⦌𝐵 ∈ V)
16	1	eleq1d 2298	. . . . 5 ⊢ (𝑦 = 𝐶 → (⦋𝑦 / 𝑥⦌𝐵 ∈ V ↔ ⦋𝐶 / 𝑥⦌𝐵 ∈ V))
17	16	spcgv 2891	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (∀𝑦⦋𝑦 / 𝑥⦌𝐵 ∈ V → ⦋𝐶 / 𝑥⦌𝐵 ∈ V))
18	15, 17	syl5 32	. . 3 ⊢ (𝐶 ∈ 𝑊 → (∀𝑥 𝐵 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌𝐵 ∈ V))
19	18	impcom 125	. 2 ⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⦋𝐶 / 𝑥⦌𝐵 ∈ V)
20		ssexg 4226	. 2 ⊢ (((𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵 ∧ ⦋𝐶 / 𝑥⦌𝐵 ∈ V) → (𝐹‘𝐶) ∈ V)
21	8, 19, 20	sylancr 414	1 ⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘𝐶) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1393   = wceq 1395   ∈ wcel 2200  Vcvv 2800  ⦋csb 3125   ⊆ wss 3198   ↦ cmpt 4148  ‘cfv 5324

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 4205  ax-pow 4262  ax-pr 4297

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 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-iota 5284  df-fun 5326  df-fv 5332

This theorem is referenced by:  mpofvex  6365  xpcomco  7005  lssex  14361  mopnset  14559  metuex  14562