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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 5568 | . 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶) | |
2 | fvmptss2.2 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funmpt2 5293 | . . . . 5 ⊢ Fun 𝐹 |
4 | funrel 5271 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ Rel 𝐹 |
6 | 5 | brrelex1i 4702 | . . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V) |
7 | nfcv 2336 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
8 | nfmpt1 4122 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 2, 8 | nfcxfr 2333 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
10 | nfcv 2336 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
11 | 7, 9, 10 | nfbr 4075 | . . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦 |
12 | nfv 1539 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶 | |
13 | 11, 12 | nfim 1583 | . . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
14 | breq1 4032 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦)) | |
15 | fvmptss2.1 | . . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
16 | 15 | sseq2d 3209 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶)) |
17 | 14, 16 | imbi12d 234 | . . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))) |
18 | df-br 4030 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
19 | opabid 4286 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) | |
20 | eqimss 3233 | . . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 ⊆ 𝐵) | |
21 | 20 | adantl 277 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ⊆ 𝐵) |
22 | 19, 21 | sylbi 121 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} → 𝑦 ⊆ 𝐵) |
23 | df-mpt 4092 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
24 | 2, 23 | eqtri 2214 | . . . . . 6 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
25 | 22, 24 | eleq2s 2288 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐹 → 𝑦 ⊆ 𝐵) |
26 | 18, 25 | sylbi 121 | . . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) |
27 | 7, 13, 17, 26 | vtoclgf 2818 | . . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)) |
28 | 6, 27 | mpcom 36 | . 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
29 | 1, 28 | mpg 1462 | 1 ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 Vcvv 2760 ⊆ wss 3153 〈cop 3621 class class class wbr 4029 {copab 4089 ↦ cmpt 4090 Rel wrel 4664 Fun wfun 5248 ‘cfv 5254 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-iota 5215 df-fun 5256 df-fv 5262 |
This theorem is referenced by: mptfvex 5643 |
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