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Mirrors > Home > ILE Home > Th. List > fvmptss2 | GIF version |
Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fvmptss2.1 | ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) |
fvmptss2.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmptss2 | ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvss 5443 | . 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶) | |
2 | fvmptss2.2 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funmpt2 5170 | . . . . 5 ⊢ Fun 𝐹 |
4 | funrel 5148 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ Rel 𝐹 |
6 | 5 | brrelex1i 4590 | . . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V) |
7 | nfcv 2282 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
8 | nfmpt1 4029 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 2, 8 | nfcxfr 2279 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
10 | nfcv 2282 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
11 | 7, 9, 10 | nfbr 3982 | . . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦 |
12 | nfv 1509 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶 | |
13 | 11, 12 | nfim 1552 | . . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
14 | breq1 3940 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦)) | |
15 | fvmptss2.1 | . . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
16 | 15 | sseq2d 3132 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶)) |
17 | 14, 16 | imbi12d 233 | . . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))) |
18 | df-br 3938 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
19 | opabid 4187 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) | |
20 | eqimss 3156 | . . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 ⊆ 𝐵) | |
21 | 20 | adantl 275 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ⊆ 𝐵) |
22 | 19, 21 | sylbi 120 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} → 𝑦 ⊆ 𝐵) |
23 | df-mpt 3999 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
24 | 2, 23 | eqtri 2161 | . . . . . 6 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
25 | 22, 24 | eleq2s 2235 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐹 → 𝑦 ⊆ 𝐵) |
26 | 18, 25 | sylbi 120 | . . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) |
27 | 7, 13, 17, 26 | vtoclgf 2747 | . . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)) |
28 | 6, 27 | mpcom 36 | . 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
29 | 1, 28 | mpg 1428 | 1 ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 〈cop 3535 class class class wbr 3937 {copab 3996 ↦ cmpt 3997 Rel wrel 4552 Fun wfun 5125 ‘cfv 5131 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-iota 5096 df-fun 5133 df-fv 5139 |
This theorem is referenced by: mptfvex 5514 |
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