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Mirrors > Home > ILE Home > Th. List > mptfvex | GIF version |
Description: Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fvmpt2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
mptfvex | ⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘𝐶) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3072 | . . 3 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) | |
2 | fvmpt2.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | nfcv 2329 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3102 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3078 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 4110 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 2, 6 | eqtri 2208 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 1, 7 | fvmptss2 5604 | . 2 ⊢ (𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵 |
9 | elex 2760 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
10 | 9 | alimi 1465 | . . . . 5 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑥 𝐵 ∈ V) |
11 | 3 | nfel1 2340 | . . . . . 6 ⊢ Ⅎ𝑦 𝐵 ∈ V |
12 | 4 | nfel1 2340 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ V |
13 | 5 | eleq1d 2256 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ V)) |
14 | 11, 12, 13 | cbval 1764 | . . . . 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 2256 | . . . . 5 ⊢ (𝑦 = 𝐶 → (⦋𝑦 / 𝑥⦌𝐵 ∈ V ↔ ⦋𝐶 / 𝑥⦌𝐵 ∈ V)) |
17 | 16 | spcgv 2836 | . . . 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 4154 | . 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) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1361 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ⦋csb 3069 ⊆ wss 3141 ↦ cmpt 4076 ‘cfv 5228 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-iota 5190 df-fun 5230 df-fv 5236 |
This theorem is referenced by: mpofvex 6218 xpcomco 6840 lssex 13543 |
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