Theorem setrec1lem3 50615

Description: Lemma for setrec1 50617. 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 50614. 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 50613 would be any easier than the current proof using setrec1lem2 50614, and it would only slightly simplify the proof of setrec1 50617. Other than the use of bnd2d 50607, this is a purely technical theorem for rearranging notation from that of setrec1lem2 50614 to that of setrec1 50617. (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 1894	. . . . . . 7 ⊢ (∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
4	3	ralbii 3108	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
5	2, 4	sylib 221	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
6		df-rex 3087	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
7	6	ralbii 3108	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
8	5, 7	sylibr 237	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥)
9	1, 8	bnd2d 50607	. . 3 ⊢ (𝜑 → ∃𝑣(𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥))
10		exancom 1894	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥) ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
11		df-rex 3087	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥))
12		eluni 4870	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑣 ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
13	10, 11, 12	3bitr4i 306	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝑎 ∈ ∪ 𝑣)
14	13	ralbii 3108	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
15		dfss3 3920	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝑣 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
16	14, 15	bitr4i 281	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣)
17	16	anbi2i 635	. . . 4 ⊢ ((𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥) ↔ (𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
18	17	exbii 1881	. . 3 ⊢ (∃𝑣(𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥) ↔ ∃𝑣(𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
19	9, 18	sylib 221	. 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 3454	. . . . . . . 8 ⊢ 𝑣 ∈ V
22	21	a1i 11	. . . . . . 7 ⊢ (𝑣 ⊆ 𝑌 → 𝑣 ∈ V)
23		id 23	. . . . . . 7 ⊢ (𝑣 ⊆ 𝑌 → 𝑣 ⊆ 𝑌)
24	20, 22, 23	setrec1lem2 50614	. . . . . 6 ⊢ (𝑣 ⊆ 𝑌 → ∪ 𝑣 ∈ 𝑌)
25	24	anim1i 627	. . . . 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 7743	. . . . 5 ⊢ ∪ 𝑣 ∈ V
28		sseq2 3957	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣))
29		eleq1 2848	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝑥 ∈ 𝑌 ↔ ∪ 𝑣 ∈ 𝑌))
30	28, 29	anbi12d 644	. . . . 5 ⊢ (𝑥 = ∪ 𝑣 → ((𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ (𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌)))
31	27, 30	spcev 3560	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))
32	26, 31	syl 18	. . 3 ⊢ ((𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))
33	32	exlimiv 1963	. 2 ⊢ (∃𝑣(𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))
34	19, 33	syl 18	1 ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  ∪ cuni 4867  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-reg 9564  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-r1 9746  df-rank 9747  df-scott 9868

This theorem is used by:  setrec1  50617