Theorem setrec1lem3 44272

Description: Lemma for setrec1 44274. 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 44271. 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 44270 would be any easier than the current proof using setrec1lem2 44271, and it would only slightly simplify the proof of setrec1 44274. Other than the use of bnd2d 44264, this is a purely technical theorem for rearranging notation from that of setrec1lem2 44271 to that of setrec1 44274. (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 1842	. . . . . . 7 ⊢ (∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
4	3	ralbii 3132	. . . . . 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 3111	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑌 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝑥))
7	6	ralbii 3132	. . . . 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 44264	. . 3 ⊢ (𝜑 → ∃𝑣(𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥))
10		exancom 1842	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥) ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
11		df-rex 3111	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝑎 ∈ 𝑥))
12		eluni 4748	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑣 ↔ ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑣))
13	10, 11, 12	3bitr4i 304	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝑎 ∈ ∪ 𝑣)
14	13	ralbii 3132	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
15		dfss3 3878	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝑣 ↔ ∀𝑎 ∈ 𝐴 𝑎 ∈ ∪ 𝑣)
16	14, 15	bitr4i 279	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣)
17	16	anbi2i 622	. . . 4 ⊢ ((𝑣 ⊆ 𝑌 ∧ ∀𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝑣 𝑎 ∈ 𝑥) ↔ (𝑣 ⊆ 𝑌 ∧ 𝐴 ⊆ ∪ 𝑣))
18	17	exbii 1829	. . 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 3440	. . . . . . . 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 44271	. . . . . 6 ⊢ (𝑣 ⊆ 𝑌 → ∪ 𝑣 ∈ 𝑌)
25	24	anim1i 614	. . . . 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 7323	. . . . 5 ⊢ ∪ 𝑣 ∈ V
28		sseq2 3914	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑣))
29		eleq1 2870	. . . . . 6 ⊢ (𝑥 = ∪ 𝑣 → (𝑥 ∈ 𝑌 ↔ ∪ 𝑣 ∈ 𝑌))
30	28, 29	anbi12d 630	. . . . 5 ⊢ (𝑥 = ∪ 𝑣 → ((𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌) ↔ (𝐴 ⊆ ∪ 𝑣 ∧ ∪ 𝑣 ∈ 𝑌)))
31	27, 30	spcev 3549	. . . 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 1908	. 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 1520   = wceq 1522  ∃wex 1761   ∈ wcel 2081  {cab 2775  ∀wral 3105  ∃wrex 3106  Vcvv 3437   ⊆ wss 3859  ∪ cuni 4745  ‘cfv 6225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-reg 8902  ax-inf2 8950

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-r1 9039  df-rank 9040

This theorem is referenced by:  setrec1  44274