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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 2911 | . . 3 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) | |
2 | fvmpt2.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | nfcv 2219 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 2938 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 2916 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 3872 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 2, 6 | eqtri 2101 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 1, 7 | fvmptss2 5268 | . 2 ⊢ (𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵 |
9 | elex 2610 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
10 | 9 | alimi 1384 | . . . . 5 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑥 𝐵 ∈ V) |
11 | 3 | nfel1 2229 | . . . . . 6 ⊢ Ⅎ𝑦 𝐵 ∈ V |
12 | 4 | nfel1 2229 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ V |
13 | 5 | eleq1d 2147 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ V)) |
14 | 11, 12, 13 | cbval 1677 | . . . . 5 ⊢ (∀𝑥 𝐵 ∈ V ↔ ∀𝑦⦋𝑦 / 𝑥⦌𝐵 ∈ V) |
15 | 10, 14 | sylib 120 | . . . 4 ⊢ (∀𝑥 𝐵 ∈ 𝑉 → ∀𝑦⦋𝑦 / 𝑥⦌𝐵 ∈ V) |
16 | 1 | eleq1d 2147 | . . . . 5 ⊢ (𝑦 = 𝐶 → (⦋𝑦 / 𝑥⦌𝐵 ∈ V ↔ ⦋𝐶 / 𝑥⦌𝐵 ∈ V)) |
17 | 16 | spcgv 2685 | . . . 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 123 | . 2 ⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⦋𝐶 / 𝑥⦌𝐵 ∈ V) |
20 | ssexg 3917 | . 2 ⊢ (((𝐹‘𝐶) ⊆ ⦋𝐶 / 𝑥⦌𝐵 ∧ ⦋𝐶 / 𝑥⦌𝐵 ∈ V) → (𝐹‘𝐶) ∈ V) | |
21 | 8, 19, 20 | sylancr 405 | 1 ⊢ ((∀𝑥 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘𝐶) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⦋csb 2908 ⊆ wss 2973 ↦ cmpt 3839 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-iota 4887 df-fun 4924 df-fv 4930 |
This theorem is referenced by: mpt2fvex 5849 xpcomco 6323 |
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