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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 2849 | . . 3 ⊢ (y = 𝐶 → ⦋y / x⦌B = ⦋𝐶 / x⦌B) | |
2 | fvmpt2.1 | . . . 4 ⊢ 𝐹 = (x ∈ A ↦ B) | |
3 | nfcv 2175 | . . . . 5 ⊢ ℲyB | |
4 | nfcsb1v 2876 | . . . . 5 ⊢ Ⅎx⦋y / x⦌B | |
5 | csbeq1a 2854 | . . . . 5 ⊢ (x = y → B = ⦋y / x⦌B) | |
6 | 3, 4, 5 | cbvmpt 3842 | . . . 4 ⊢ (x ∈ A ↦ B) = (y ∈ A ↦ ⦋y / x⦌B) |
7 | 2, 6 | eqtri 2057 | . . 3 ⊢ 𝐹 = (y ∈ A ↦ ⦋y / x⦌B) |
8 | 1, 7 | fvmptss2 5190 | . 2 ⊢ (𝐹‘𝐶) ⊆ ⦋𝐶 / x⦌B |
9 | elex 2560 | . . . . . 6 ⊢ (B ∈ 𝑉 → B ∈ V) | |
10 | 9 | alimi 1341 | . . . . 5 ⊢ (∀x B ∈ 𝑉 → ∀x B ∈ V) |
11 | 3 | nfel1 2185 | . . . . . 6 ⊢ Ⅎy B ∈ V |
12 | 4 | nfel1 2185 | . . . . . 6 ⊢ Ⅎx⦋y / x⦌B ∈ V |
13 | 5 | eleq1d 2103 | . . . . . 6 ⊢ (x = y → (B ∈ V ↔ ⦋y / x⦌B ∈ V)) |
14 | 11, 12, 13 | cbval 1634 | . . . . 5 ⊢ (∀x B ∈ V ↔ ∀y⦋y / x⦌B ∈ V) |
15 | 10, 14 | sylib 127 | . . . 4 ⊢ (∀x B ∈ 𝑉 → ∀y⦋y / x⦌B ∈ V) |
16 | 1 | eleq1d 2103 | . . . . 5 ⊢ (y = 𝐶 → (⦋y / x⦌B ∈ V ↔ ⦋𝐶 / x⦌B ∈ V)) |
17 | 16 | spcgv 2634 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (∀y⦋y / x⦌B ∈ V → ⦋𝐶 / x⦌B ∈ V)) |
18 | 15, 17 | syl5 28 | . . 3 ⊢ (𝐶 ∈ 𝑊 → (∀x B ∈ 𝑉 → ⦋𝐶 / x⦌B ∈ V)) |
19 | 18 | impcom 116 | . 2 ⊢ ((∀x B ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⦋𝐶 / x⦌B ∈ V) |
20 | ssexg 3887 | . 2 ⊢ (((𝐹‘𝐶) ⊆ ⦋𝐶 / x⦌B ∧ ⦋𝐶 / x⦌B ∈ V) → (𝐹‘𝐶) ∈ V) | |
21 | 8, 19, 20 | sylancr 393 | 1 ⊢ ((∀x B ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹‘𝐶) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ⦋csb 2846 ⊆ wss 2911 ↦ cmpt 3809 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-iota 4810 df-fun 4847 df-fv 4853 |
This theorem is referenced by: mpt2fvex 5771 xpcomco 6236 |
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