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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1147 | Structured version Visualization version GIF version |
Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1147 | ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid 263 | . 2 ⊢ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
2 | biid 263 | . 2 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
3 | eqid 2821 | . 2 ⊢ (ω ∖ {∅}) = (ω ∖ {∅}) | |
4 | eqid 2821 | . 2 ⊢ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))} = {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))} | |
5 | biid 263 | . 2 ⊢ ((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) | |
6 | biid 263 | . 2 ⊢ (((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ (𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ (𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) | |
7 | 1, 2, 3, 4, 5, 6 | bnj1145 32265 | 1 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 {csn 4567 ∪ ciun 4919 suc csuc 6193 Fn wfn 6350 ‘cfv 6355 ωcom 7580 ∧ w-bnj17 31956 predc-bnj14 31958 trClc-bnj18 31964 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fn 6358 df-fv 6363 df-om 7581 df-bnj17 31957 df-bnj14 31959 df-bnj18 31965 |
This theorem is referenced by: bnj1137 32267 bnj1173 32274 bnj1177 32278 bnj1408 32308 bnj1417 32313 bnj1452 32324 |
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