Theorem setrec1lem3 42436

Description: Lemma for setrec1 42438. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 42435. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴 ∈ 𝑌. I don't know if proving this fact directly using setrec1lem1 42434 would be any easier than the current proof using setrec1lem2 42435, and it would only slightly simplify the proof of setrec1 42438. Other than the use of bnd2d 42428, this is a purely technical theorem for rearranging notation from that of setrec1lem2 42435 to that of setrec1 42438. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.)

Hypotheses

Ref	Expression
setrec1lem3.1	⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
setrec1lem3.2	⊢ (𝜑 → 𝐴 ∈ V)
setrec1lem3.3	⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌))

Assertion

Ref	Expression
setrec1lem3	⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))

Distinct variable groups:   𝑦,𝑤,𝑧   𝑥,𝑎,𝐴   𝑌,𝑎,𝑥   𝑥,𝑦,𝐹

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑎)   𝐴(𝑦,𝑧,𝑤)   𝐹(𝑧,𝑤,𝑎)   𝑌(𝑦,𝑧,𝑤)

Proof of Theorem setrec1lem3

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setrec1lem3.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
2		setrec1lem3.3	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌))
3		exancom 1787	. . . . . . 7 ⊢ (∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
4	3	ralbii 2980	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
5	2, 4	sylib 208	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
6		df-rex 2918	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
7	6	ralbii 2980	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
8	5, 7	sylibr 224	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥)
9	1, 8	bnd2d 42428	. . 3 ⊢ (𝜑 → ∃𝑣(𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥))
10		exancom 1787	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥) ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
11		df-rex 2918	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥))
12		eluni 4439	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑣 ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
13	10, 11, 12	3bitr4i 292	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝑎 ∈ ∪ 𝑣)
14	13	ralbii 2980	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
15		dfss3 3592	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝑣 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
16	14, 15	bitr4i 267	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣)
17	16	anbi2i 730	. . . 4 ⊢ ((𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥) ↔ (𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
18	17	exbii 1774	. . 3 ⊢ (∃𝑣(𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥) ↔ ∃𝑣(𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
19	9, 18	sylib 208	. 2 ⊢ (𝜑 → ∃𝑣(𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
20		setrec1lem3.1	. . . . . . 7 ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
21		vex 3203	. . . . . . . 8 ⊢ 𝑣 ∈ V
22	21	a1i 11	. . . . . . 7 ⊢ (𝑣 ⊆ 𝑌 → 𝑣 ∈ V)
23		id 22	. . . . . . 7 ⊢ (𝑣 ⊆ 𝑌 → 𝑣 ⊆ 𝑌)
24	20, 22, 23	setrec1lem2 42435	. . . . . 6 ⊢ (𝑣 ⊆ 𝑌 → ∪ 𝑣 ∈ 𝑌)
25	24	anim1i 592	. . . . 5 ⊢ ((𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → (∪ 𝑣 ∈ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
26	25	ancomd 467	. . . 4 ⊢ ((𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → (𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌))
27	21	uniex 6953	. . . . 5 ⊢ ∪ 𝑣 ∈ V
28		sseq2 3627	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣))
29		eleq1 2689	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝑥 ∈ 𝑌 ↔ ∪ 𝑣 ∈ 𝑌))
30	28, 29	anbi12d 747	. . . . 5 ⊢ (𝑥 = ∪ 𝑣 → ((𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ (𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌)))
31	27, 30	spcev 3300	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))
32	26, 31	syl 17	. . 3 ⊢ ((𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))
33	32	exlimiv 1858	. 2 ⊢ (∃𝑣(𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))
34	19, 33	syl 17	1 ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∪ cuni 4436  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  setrec1  42438