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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 5529 | . 2 ⊢ (∀𝑦(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶) | |
2 | fvmptss2.2 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | funmpt2 5255 | . . . . 5 ⊢ Fun 𝐹 |
4 | funrel 5233 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ Rel 𝐹 |
6 | 5 | brrelex1i 4669 | . . 3 ⊢ (𝐷𝐹𝑦 → 𝐷 ∈ V) |
7 | nfcv 2319 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
8 | nfmpt1 4096 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 2, 8 | nfcxfr 2316 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
10 | nfcv 2319 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
11 | 7, 9, 10 | nfbr 4049 | . . . . 5 ⊢ Ⅎ𝑥 𝐷𝐹𝑦 |
12 | nfv 1528 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐶 | |
13 | 11, 12 | nfim 1572 | . . . 4 ⊢ Ⅎ𝑥(𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
14 | breq1 4006 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝑦 ↔ 𝐷𝐹𝑦)) | |
15 | fvmptss2.1 | . . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
16 | 15 | sseq2d 3185 | . . . . 5 ⊢ (𝑥 = 𝐷 → (𝑦 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐶)) |
17 | 14, 16 | imbi12d 234 | . . . 4 ⊢ (𝑥 = 𝐷 → ((𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) ↔ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶))) |
18 | df-br 4004 | . . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
19 | opabid 4257 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) | |
20 | eqimss 3209 | . . . . . . . 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 4066 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
24 | 2, 23 | eqtri 2198 | . . . . . 6 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
25 | 22, 24 | eleq2s 2272 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐹 → 𝑦 ⊆ 𝐵) |
26 | 18, 25 | sylbi 121 | . . . 4 ⊢ (𝑥𝐹𝑦 → 𝑦 ⊆ 𝐵) |
27 | 7, 13, 17, 26 | vtoclgf 2795 | . . 3 ⊢ (𝐷 ∈ V → (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶)) |
28 | 6, 27 | mpcom 36 | . 2 ⊢ (𝐷𝐹𝑦 → 𝑦 ⊆ 𝐶) |
29 | 1, 28 | mpg 1451 | 1 ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 〈cop 3595 class class class wbr 4003 {copab 4063 ↦ cmpt 4064 Rel wrel 4631 Fun wfun 5210 ‘cfv 5216 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-iota 5178 df-fun 5218 df-fv 5224 |
This theorem is referenced by: mptfvex 5601 |
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