Theorem setrec1lem3 50179

Description: Lemma for setrec1 50181. 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 50178. 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 50177 would be any easier than the current proof using setrec1lem2 50178, and it would only slightly simplify the proof of setrec1 50181. Other than the use of bnd2d 50171, this is a purely technical theorem for rearranging notation from that of setrec1lem2 50178 to that of setrec1 50181. (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 1868	. . . . . . 7 ⊢ (∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
4	3	ralbii 3085	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
5	2, 4	sylib 219	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
6		df-rex 3064	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
7	6	ralbii 3085	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
8	5, 7	sylibr 235	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥)
9	1, 8	bnd2d 50171	. . 3 ⊢ (𝜑 → ∃𝑣(𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥))
10		exancom 1868	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥) ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
11		df-rex 3064	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥))
12		eluni 4841	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑣 ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
13	10, 11, 12	3bitr4i 304	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝑎 ∈ ∪ 𝑣)
14	13	ralbii 3085	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
15		dfss3 3904	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝑣 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
16	14, 15	bitr4i 279	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣)
17	16	anbi2i 629	. . . 4 ⊢ ((𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥) ↔ (𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
18	17	exbii 1855	. . 3 ⊢ (∃𝑣(𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥) ↔ ∃𝑣(𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
19	9, 18	sylib 219	. 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 3435	. . . . . . . 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 50178	. . . . . 6 ⊢ (𝑣 ⊆ 𝑌 → ∪ 𝑣 ∈ 𝑌)
25	24	anim1i 621	. . . . 5 ⊢ ((𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → (∪ 𝑣 ∈ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
26	25	ancomd 462	. . . 4 ⊢ ((𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → (𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌))
27	21	uniex 7684	. . . . 5 ⊢ ∪ 𝑣 ∈ V
28		sseq2 3941	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣))
29		eleq1 2827	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝑥 ∈ 𝑌 ↔ ∪ 𝑣 ∈ 𝑌))
30	28, 29	anbi12d 638	. . . . 5 ⊢ (𝑥 = ∪ 𝑣 → ((𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ (𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌)))
31	27, 30	spcev 3544	. . . 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 1937	. 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 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∪ cuni 4838  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-reg 9497  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9679  df-rank 9680

This theorem is referenced by:  setrec1  50181