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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 3052 | . . 3 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) | |
2 | fvmpt2.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | nfcv 2312 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3082 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3058 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 4082 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 2, 6 | eqtri 2191 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 1, 7 | fvmptss2 5569 | . 2 ⊢ (𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵 |
9 | elex 2741 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
10 | 9 | alimi 1448 | . . . . 5 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑥 𝐵 ∈ V) |
11 | 3 | nfel1 2323 | . . . . . 6 ⊢ Ⅎ𝑦 𝐵 ∈ V |
12 | 4 | nfel1 2323 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ V |
13 | 5 | eleq1d 2239 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ V)) |
14 | 11, 12, 13 | cbval 1747 | . . . . 5 ⊢ (∀𝑥 𝐵 ∈ V ↔ ∀𝑦⦋𝑦 / 𝑥⦌𝐵 ∈ V) |
15 | 10, 14 | sylib 121 | . . . 4 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑦⦋𝑦 / 𝑥⦌𝐵 ∈ V) |
16 | 1 | eleq1d 2239 | . . . . 5 ⊢ (𝑦 = 𝐶 → (⦋𝑦 / 𝑥⦌𝐵 ∈ V ↔ ⦋𝐶 / 𝑥⦌𝐵 ∈ V)) |
17 | 16 | spcgv 2817 | . . . 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 124 | . 2 ⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⦋𝐶 / 𝑥⦌𝐵 ∈ V) |
20 | ssexg 4126 | . 2 ⊢ (((𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵 ∧ ⦋𝐶 / 𝑥⦌𝐵 ∈ V) → (𝐹‘𝐶) ∈ V) | |
21 | 8, 19, 20 | sylancr 412 | 1 ⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘𝐶) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1346 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ⦋csb 3049 ⊆ wss 3121 ↦ cmpt 4048 ‘cfv 5196 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-iota 5158 df-fun 5198 df-fv 5204 |
This theorem is referenced by: mpofvex 6180 xpcomco 6802 |
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