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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 2826 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
| 3 | fvmptn.2 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 4 | 3 | dmmpt 6260 | . . . 4 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
| 5 | 2, 4 | elrab2 3695 | . . 3 ⊢ (𝐷 ∈ dom 𝐹 ↔ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V)) |
| 6 | 1, 3 | fvmptg 7014 | . . . 4 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) = 𝐶) |
| 7 | eqimss 4042 | . . . 4 ⊢ ((𝐹‘𝐷) = 𝐶 → (𝐹‘𝐷) ⊆ 𝐶) | |
| 8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) ⊆ 𝐶) |
| 9 | 5, 8 | sylbi 217 | . 2 ⊢ (𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶) |
| 10 | ndmfv 6941 | . . 3 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅) | |
| 11 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ 𝐶 | |
| 12 | 10, 11 | eqsstrdi 4028 | . 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: cvmsi 35270 |
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