Theorem elsetrecslem 43560

Description: Lemma for elsetrecs 43561. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 43558. To see why this lemma also requires setrec1 43553, 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 4551	. . . . 5 ⊢ (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵))
2	1	simprbi 492	. . . 4 ⊢ (𝐵 ⊆ (𝐵 ∖ {𝐴}) → ¬ 𝐴 ∈ 𝐵)
3	2	con2i 137	. . 3 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 ⊆ (𝐵 ∖ {𝐴}))
4		elsetrecs.1	. . . 4 ⊢ 𝐵 = setrecs(𝐹)
5		sseq1 3845	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐵 ↔ 𝑎 ⊆ 𝐵))
6		fveq2 6448	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
7	6	eleq2d 2845	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ (𝐹‘𝑎)))
8	5, 7	anbi12d 624	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))))
9	8	notbid 310	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))))
10	9	spv 2358	. . . . . 6 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)))
11		imnan 390	. . . . . . . . 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 3401	. . . . . . . . . . . . 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 43553	. . . . . . . . . . 11 ⊢ (𝑎 ⊆ 𝐵 → (𝐹‘𝑎) ⊆ 𝐵)
17	12, 16	jctild 521	. . . . . . . . . 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 227	. . . . . . . 8 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → (𝑎 ⊆ 𝐵 → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎))))
20	19	adantrd 487	. . . . . . 7 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → ((𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎) → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎))))
21		ssdifsn 4551	. . . . . . 7 ⊢ (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎))
22		ssdifsn 4551	. . . . . . 7 ⊢ ((𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎)))
23	20, 21, 22	3imtr4g 288	. . . . . 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 1970	. . . 4 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴})))
26	4, 25	setrec2v 43558	. . 3 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐵 ⊆ (𝐵 ∖ {𝐴}))
27	3, 26	nsyl 138	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ ∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
28		df-ex 1824	. 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
29	27, 28	sylibr 226	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2107  Vcvv 3398   ∖ cdif 3789   ⊆ wss 3792  {csn 4398  ‘cfv 6137  setrecscsetrecs 43545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-reg 8788  ax-inf2 8837

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-r1 8926  df-rank 8927  df-setrecs 43546

This theorem is referenced by:  elsetrecs  43561