Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fvmptss2 | Structured version Visualization version 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.) |
Ref | Expression |
---|---|
fvmptn.1 | ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) |
fvmptn.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmptss2 | ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptn.1 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
2 | 1 | eleq1d 2823 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
3 | fvmptn.2 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmpt 6143 | . . . 4 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | 2, 4 | elrab2 3627 | . . 3 ⊢ (𝐷 ∈ dom 𝐹 ↔ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V)) |
6 | 1, 3 | fvmptg 6873 | . . . 4 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) = 𝐶) |
7 | eqimss 3977 | . . . 4 ⊢ ((𝐹‘𝐷) = 𝐶 → (𝐹‘𝐷) ⊆ 𝐶) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) ⊆ 𝐶) |
9 | 5, 8 | sylbi 216 | . 2 ⊢ (𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶) |
10 | ndmfv 6804 | . . 3 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅) | |
11 | 0ss 4330 | . . 3 ⊢ ∅ ⊆ 𝐶 | |
12 | 10, 11 | eqsstrdi 3975 | . 2 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶) |
13 | 9, 12 | pm2.61i 182 | 1 ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 ↦ cmpt 5157 dom cdm 5589 ‘cfv 6433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fv 6441 |
This theorem is referenced by: cvmsi 33227 |
Copyright terms: Public domain | W3C validator |