Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mptssALT | Structured version Visualization version GIF version |
Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5912. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mptssALT | ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3963 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | anim1d 612 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶))) |
3 | 2 | ssopab2dv 5440 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)}) |
4 | df-mpt 5149 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
5 | df-mpt 5149 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)} | |
6 | 3, 4, 5 | 3sstr4g 4014 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {copab 5130 ↦ cmpt 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-in 3945 df-ss 3954 df-opab 5131 df-mpt 5149 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |