Theorem mptfvex 5762

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 3140	. . 3 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
2		fvmpt2.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfcv 2384	. . . . 5 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3170	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3146	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 4204	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2253	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmptss2 5751	. 2 ⊢ (𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵
9		elex 2824	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
10	9	alimi 1504	. . . . 5 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑥 𝐵 ∈ V)
11	3	nfel1 2395	. . . . . 6 ⊢ Ⅎ𝑦 𝐵 ∈ V
12	4	nfel1 2395	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ V
13	5	eleq1d 2301	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ V))
14	11, 12, 13	cbval 1803	. . . . 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 2301	. . . . 5 ⊢ (𝑦 = 𝐶 → (⦋𝑦 / 𝑥⦌𝐵 ∈ V ↔ ⦋𝐶 / 𝑥⦌𝐵 ∈ V))
17	16	spcgv 2903	. . . 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 4248	. 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 1396   = wceq 1398   ∈ wcel 2203  Vcvv 2812  ⦋csb 3137   ⊆ wss 3210   ↦ cmpt 4170  ‘cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-iota 5311  df-fun 5353  df-fv 5359

This theorem is referenced by:  mpofvex  6400  xpcomco  7076  lssex  14489  mopnset  14687  metuex  14690