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Mirrors > Home > MPE Home > Th. List > Mathboxes > clrellem | Structured version Visualization version GIF version |
Description: When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.) |
Ref | Expression |
---|---|
clrellem.y | ⊢ (𝜑 → 𝑌 ∈ V) |
clrellem.rel | ⊢ (𝜑 → Rel 𝑋) |
clrellem.sub | ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒)) |
clrellem.sup | ⊢ (𝜑 → 𝑋 ⊆ 𝑌) |
clrellem.maj | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
clrellem | ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clrellem.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ V) | |
2 | cnvexg 7005 | . . . 4 ⊢ (𝑌 ∈ V → ◡𝑌 ∈ V) | |
3 | cnvexg 7005 | . . . 4 ⊢ (◡𝑌 ∈ V → ◡◡𝑌 ∈ V) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡◡𝑌 ∈ V) |
5 | clrellem.rel | . . . . . 6 ⊢ (𝜑 → Rel 𝑋) | |
6 | dfrel2 5502 | . . . . . 6 ⊢ (Rel 𝑋 ↔ ◡◡𝑋 = 𝑋) | |
7 | 5, 6 | sylib 207 | . . . . 5 ⊢ (𝜑 → ◡◡𝑋 = 𝑋) |
8 | clrellem.sup | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌) | |
9 | cnvss 5216 | . . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → ◡𝑋 ⊆ ◡𝑌) | |
10 | cnvss 5216 | . . . . . 6 ⊢ (◡𝑋 ⊆ ◡𝑌 → ◡◡𝑋 ⊆ ◡◡𝑌) | |
11 | 8, 9, 10 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ◡◡𝑋 ⊆ ◡◡𝑌) |
12 | 7, 11 | eqsstr3d 3603 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ◡◡𝑌) |
13 | clrellem.maj | . . . 4 ⊢ (𝜑 → 𝜒) | |
14 | relcnv 5422 | . . . . 5 ⊢ Rel ◡◡𝑌 | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → Rel ◡◡𝑌) |
16 | 12, 13, 15 | jca31 555 | . . 3 ⊢ (𝜑 → ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌)) |
17 | clrellem.sub | . . . . 5 ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒)) | |
18 | 17 | cleq2lem 36933 | . . . 4 ⊢ (𝑥 = ◡◡𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ ◡◡𝑌 ∧ 𝜒))) |
19 | releq 5124 | . . . 4 ⊢ (𝑥 = ◡◡𝑌 → (Rel 𝑥 ↔ Rel ◡◡𝑌)) | |
20 | 18, 19 | anbi12d 743 | . . 3 ⊢ (𝑥 = ◡◡𝑌 → (((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) ↔ ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌))) |
21 | 4, 16, 20 | elabd 3321 | . 2 ⊢ (𝜑 → ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥)) |
22 | releq 5124 | . . . 4 ⊢ (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥)) | |
23 | 22 | rexab2 3340 | . . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥)) |
24 | 23 | biimpri 217 | . 2 ⊢ (∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦) |
25 | relint 5165 | . 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) | |
26 | 21, 24, 25 | 3syl 18 | 1 ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ∩ cint 4410 ◡ccnv 5037 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iin 4458 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: (None) |
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