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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 5189 | . 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶) | |
2 | fvmptss2.2 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funmpt2 4939 | . . . . 5 ⊢ Fun 𝐹 |
4 | funrel 4919 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 7 | . . . 4 ⊢ Rel 𝐹 |
6 | 5 | brrelexi 4384 | . . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V) |
7 | nfcv 2178 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
8 | nfmpt1 3850 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 2, 8 | nfcxfr 2175 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
10 | nfcv 2178 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
11 | 7, 9, 10 | nfbr 3808 | . . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦 |
12 | nfv 1421 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶 | |
13 | 11, 12 | nfim 1464 | . . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
14 | breq1 3767 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦)) | |
15 | fvmptss2.1 | . . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
16 | 15 | sseq2d 2973 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶)) |
17 | 14, 16 | imbi12d 223 | . . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))) |
18 | df-br 3765 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
19 | opabid 3994 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) | |
20 | eqimss 2997 | . . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 ⊆ 𝐵) | |
21 | 20 | adantl 262 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ⊆ 𝐵) |
22 | 19, 21 | sylbi 114 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} → 𝑦 ⊆ 𝐵) |
23 | df-mpt 3820 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
24 | 2, 23 | eqtri 2060 | . . . . . 6 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
25 | 22, 24 | eleq2s 2132 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐹 → 𝑦 ⊆ 𝐵) |
26 | 18, 25 | sylbi 114 | . . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) |
27 | 7, 13, 17, 26 | vtoclgf 2612 | . . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)) |
28 | 6, 27 | mpcom 32 | . 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
29 | 1, 28 | mpg 1340 | 1 ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 〈cop 3378 class class class wbr 3764 {copab 3817 ↦ cmpt 3818 Rel wrel 4350 Fun wfun 4896 ‘cfv 4902 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-iota 4867 df-fun 4904 df-fv 4910 |
This theorem is referenced by: mptfvex 5256 |
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