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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 2874 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
3 | fvmptn.2 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmpt 6061 | . . . 4 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | 2, 4 | elrab2 3631 | . . 3 ⊢ (𝐷 ∈ dom 𝐹 ↔ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V)) |
6 | 1, 3 | fvmptg 6743 | . . . 4 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) = 𝐶) |
7 | eqimss 3971 | . . . 4 ⊢ ((𝐹‘𝐷) = 𝐶 → (𝐹‘𝐷) ⊆ 𝐶) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) ⊆ 𝐶) |
9 | 5, 8 | sylbi 220 | . 2 ⊢ (𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶) |
10 | ndmfv 6675 | . . 3 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅) | |
11 | 0ss 4304 | . . 3 ⊢ ∅ ⊆ 𝐶 | |
12 | 10, 11 | eqsstrdi 3969 | . 2 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶) |
13 | 9, 12 | pm2.61i 185 | 1 ⊢ (𝐹‘𝐷) ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: cvmsi 32625 |
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