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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsetrecslem | Structured version Visualization version GIF version |
Description: Lemma for elsetrecs 47832. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 47828. To see why this lemma also requires setrec1 47823, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elsetrecs.1 | ⊢ 𝐵 = setrecs(𝐹) |
Ref | Expression |
---|---|
elsetrecslem | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdifsn 4790 | . . . . 5 ⊢ (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵)) | |
2 | 1 | simprbi 495 | . . . 4 ⊢ (𝐵 ⊆ (𝐵 ∖ {𝐴}) → ¬ 𝐴 ∈ 𝐵) |
3 | 2 | con2i 139 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 ⊆ (𝐵 ∖ {𝐴})) |
4 | elsetrecs.1 | . . . 4 ⊢ 𝐵 = setrecs(𝐹) | |
5 | sseq1 4006 | . . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐵 ↔ 𝑎 ⊆ 𝐵)) | |
6 | fveq2 6890 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) | |
7 | 6 | eleq2d 2817 | . . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ (𝐹‘𝑎))) |
8 | 5, 7 | anbi12d 629 | . . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)))) |
9 | 8 | notbid 317 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)))) |
10 | 9 | spvv 1998 | . . . . . 6 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))) |
11 | imnan 398 | . . . . . . . . 9 ⊢ ((𝑎 ⊆ 𝐵 → ¬ 𝐴 ∈ (𝐹‘𝑎)) ↔ ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))) | |
12 | idd 24 | . . . . . . . . . . 11 ⊢ (𝑎 ⊆ 𝐵 → (¬ 𝐴 ∈ (𝐹‘𝑎) → ¬ 𝐴 ∈ (𝐹‘𝑎))) | |
13 | vex 3476 | . . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V | |
14 | 13 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑎 ⊆ 𝐵 → 𝑎 ∈ V) |
15 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑎 ⊆ 𝐵 → 𝑎 ⊆ 𝐵) | |
16 | 4, 14, 15 | setrec1 47823 | . . . . . . . . . . 11 ⊢ (𝑎 ⊆ 𝐵 → (𝐹‘𝑎) ⊆ 𝐵) |
17 | 12, 16 | jctild 524 | . . . . . . . . . 10 ⊢ (𝑎 ⊆ 𝐵 → (¬ 𝐴 ∈ (𝐹‘𝑎) → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎)))) |
18 | 17 | a2i 14 | . . . . . . . . 9 ⊢ ((𝑎 ⊆ 𝐵 → ¬ 𝐴 ∈ (𝐹‘𝑎)) → (𝑎 ⊆ 𝐵 → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎)))) |
19 | 11, 18 | sylbir 234 | . . . . . . . 8 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → (𝑎 ⊆ 𝐵 → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎)))) |
20 | 19 | adantrd 490 | . . . . . . 7 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → ((𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎) → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎)))) |
21 | ssdifsn 4790 | . . . . . . 7 ⊢ (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎)) | |
22 | ssdifsn 4790 | . . . . . . 7 ⊢ ((𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎))) | |
23 | 20, 21, 22 | 3imtr4g 295 | . . . . . 6 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴}))) |
24 | 10, 23 | syl 17 | . . . . 5 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴}))) |
25 | 24 | alrimiv 1928 | . . . 4 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴}))) |
26 | 4, 25 | setrec2v 47828 | . . 3 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐵 ⊆ (𝐵 ∖ {𝐴})) |
27 | 3, 26 | nsyl 140 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) |
28 | df-ex 1780 | . 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | |
29 | 27, 28 | sylibr 233 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∀wal 1537 = wceq 1539 ∃wex 1779 ∈ wcel 2104 Vcvv 3472 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 ‘cfv 6542 setrecscsetrecs 47815 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-r1 9761 df-rank 9762 df-setrecs 47816 |
This theorem is referenced by: elsetrecs 47832 |
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