Theorem elsetrecslem 47832

Description: Lemma for elsetrecs 47833. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 47829. To see why this lemma also requires setrec1 47824, 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.)

Hypothesis

Ref	Expression
elsetrecs.1	⊢ 𝐵 = setrecs(𝐹)

Assertion

Ref	Expression
elsetrecslem	⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem elsetrecslem

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssdifsn 4791	. . . . 5 ⊢ (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵))
2	1	simprbi 496	. . . 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 4007	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐵 ↔ 𝑎 ⊆ 𝐵))
6		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
7	6	eleq2d 2818	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ (𝐹‘𝑎)))
8	5, 7	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))))
9	8	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))))
10	9	spvv 1999	. . . . . 6 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)))
11		imnan 399	. . . . . . . . 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 3477	. . . . . . . . . . . . 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 47824	. . . . . . . . . . 11 ⊢ (𝑎 ⊆ 𝐵 → (𝐹‘𝑎) ⊆ 𝐵)
17	12, 16	jctild 525	. . . . . . . . . 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 491	. . . . . . 7 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → ((𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎) → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎))))
21		ssdifsn 4791	. . . . . . 7 ⊢ (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎))
22		ssdifsn 4791	. . . . . . 7 ⊢ ((𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎)))
23	20, 21, 22	3imtr4g 296	. . . . . 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 1929	. . . 4 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴})))
26	4, 25	setrec2v 47829	. . 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 1781	. 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)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3473   ∖ cdif 3945   ⊆ wss 3948  {csn 4628  ‘cfv 6543  setrecscsetrecs 47816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-reg 9591  ax-inf2 9640

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-r1 9763  df-rank 9764  df-setrecs 47817

This theorem is referenced by:  elsetrecs  47833