Theorem elsetrecslem 49685

Description: Lemma for elsetrecs 49686. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 49682. To see why this lemma also requires setrec1 49677, 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 4752	. . . . 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 3972	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐵 ↔ 𝑎 ⊆ 𝐵))
6		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
7	6	eleq2d 2814	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ (𝐹‘𝑎)))
8	5, 7	anbi12d 632	. . . . . . . 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 1988	. . . . . 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 3451	. . . . . . . . . . . . 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 49677	. . . . . . . . . . 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 235	. . . . . . . 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 4752	. . . . . . 7 ⊢ (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎))
22		ssdifsn 4752	. . . . . . 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 1927	. . . 4 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴})))
26	4, 25	setrec2v 49682	. . 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 1780	. 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
29	27, 28	sylibr 234	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  {csn 4589  ‘cfv 6511  setrecscsetrecs 49669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717  df-rank 9718  df-setrecs 49670

This theorem is referenced by:  elsetrecs  49686