Theorem elsetrecslem 50325

Description: Lemma for elsetrecs 50326. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 50322. To see why this lemma also requires setrec1 50317, 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 4750	. . . . 5 ⊢ (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵))
2	1	simprbi 501	. . . 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 3963	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐵 ↔ 𝑎 ⊆ 𝐵))
6		fveq2 6869	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
7	6	eleq2d 2850	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ (𝐹‘𝑎)))
8	5, 7	anbi12d 641	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))))
9	8	notbid 320	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎))))
10	9	spvv 2010	. . . . . 6 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)))
11		imnan 403	. . . . . . . . 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 3460	. . . . . . . . . . . . 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 50317	. . . . . . . . . . 11 ⊢ (𝑎 ⊆ 𝐵 → (𝐹‘𝑎) ⊆ 𝐵)
17	12, 16	jctild 533	. . . . . . . . . 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 237	. . . . . . . 8 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → (𝑎 ⊆ 𝐵 → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎))))
20	19	adantrd 495	. . . . . . 7 ⊢ (¬ (𝑎 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑎)) → ((𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎) → ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎))))
21		ssdifsn 4750	. . . . . . 7 ⊢ (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎 ⊆ 𝐵 ∧ ¬ 𝐴 ∈ 𝑎))
22		ssdifsn 4750	. . . . . . 7 ⊢ ((𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹‘𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹‘𝑎)))
23	20, 21, 22	3imtr4g 298	. . . . . 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 1949	. . . 4 ⊢ (∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹‘𝑎) ⊆ (𝐵 ∖ {𝐴})))
26	4, 25	setrec2v 50322	. . 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 1802	. 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
29	27, 28	sylibr 236	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1560   = wceq 1562  ∃wex 1801   ∈ wcel 2144  Vcvv 3456   ∖ cdif 3903   ⊆ wss 3906  {csn 4584  ‘cfv 6523  setrecscsetrecs 50309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-reg 9542  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  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 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-r1 9724  df-rank 9725  df-setrecs 50310

This theorem is referenced by:  elsetrecs  50326