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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsetrecs | Structured version Visualization version GIF version |
Description: A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 42237 and setrec2 42241, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of ∈ are replaced by ⊆ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
Ref | Expression |
---|---|
elsetrecs.1 | ⊢ 𝐵 = setrecs(𝐹) |
Ref | Expression |
---|---|
elsetrecs | ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsetrecs.1 | . . 3 ⊢ 𝐵 = setrecs(𝐹) | |
2 | 1 | elsetrecslem 42243 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) |
3 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ∈ V) |
5 | id 22 | . . . . 5 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐵) | |
6 | 1, 4, 5 | setrec1 42237 | . . . 4 ⊢ (𝑥 ⊆ 𝐵 → (𝐹‘𝑥) ⊆ 𝐵) |
7 | 6 | sselda 3568 | . . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐴 ∈ 𝐵) |
8 | 7 | exlimiv 1845 | . 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐴 ∈ 𝐵) |
9 | 2, 8 | impbii 198 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ‘cfv 5804 setrecscsetrecs 42229 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-rank 8511 df-setrecs 42230 |
This theorem is referenced by: elpg 42256 |
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