Theorem zorn2lem1 9921

Description: Lemma for zorn2 9931. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
zorn2lem.3	⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))
zorn2lem.4	⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
zorn2lem.5	⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}

Assertion

Ref	Expression
zorn2lem1	⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)

Distinct variable groups:   𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧,𝐴   𝐷,𝑓,𝑢,𝑣   𝑓,𝐹,𝑔,𝑢,𝑣,𝑥,𝑧   𝑅,𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧   𝑣,𝐶

Allowed substitution hints:   𝐶(𝑥,𝑧,𝑤,𝑢,𝑓,𝑔)   𝐷(𝑥,𝑧,𝑤,𝑔)   𝐹(𝑤)

Proof of Theorem zorn2lem1

Step	Hyp	Ref	Expression
1		zorn2lem.3	. . . . 5 ⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))
2	1	tfr2 8037	. . . 4 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
3	2	adantr 483	. . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
4	1	tfr1 8036	. . . . . 6 ⊢ 𝐹 Fn On
5		fnfun 6456	. . . . . 6 ⊢ (𝐹 Fn On → Fun 𝐹)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
7		vex 3500	. . . . 5 ⊢ 𝑥 ∈ V
8		resfunexg 6981	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
9	6, 7, 8	mp2an 690	. . . 4 ⊢ (𝐹 ↾ 𝑥) ∈ V
10		rneq 5809	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
11		df-ima 5571	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
12	10, 11	syl6eqr 2877	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
13	12	eleq2d 2901	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑔 ∈ ran 𝑓 ↔ 𝑔 ∈ (𝐹 “ 𝑥)))
14	13	imbi1d 344	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑔 ∈ ran 𝑓 → 𝑔𝑅𝑧) ↔ (𝑔 ∈ (𝐹 “ 𝑥) → 𝑔𝑅𝑧)))
15	14	ralbidv2 3198	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧))
16	15	rabbidv 3483	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧})
17		zorn2lem.4	. . . . . . 7 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
18		zorn2lem.5	. . . . . . 7 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
19	16, 17, 18	3eqtr4g 2884	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → 𝐶 = 𝐷)
20	19	eleq2d 2901	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑢 ∈ 𝐶 ↔ 𝑢 ∈ 𝐷))
21	20	imbi1d 344	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑢 ∈ 𝐶 → ¬ 𝑢𝑤𝑣) ↔ (𝑢 ∈ 𝐷 → ¬ 𝑢𝑤𝑣)))
22	21	ralbidv2 3198	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣 ↔ ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
23	19, 22	riotaeqbidv 7120	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
24		eqid 2824	. . . . 5 ⊢ (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)) = (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))
25		riotaex 7121	. . . . 5 ⊢ (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣) ∈ V
26	23, 24, 25	fvmpt 6771	. . . 4 ⊢ ((𝐹 ↾ 𝑥) ∈ V → ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
27	9, 26	ax-mp 5	. . 3 ⊢ ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣)
28	3, 27	syl6eq 2875	. 2 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
29		simprl 769	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝑤 We 𝐴)
30		weso 5549	. . . . . . 7 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
31	30	ad2antrl 726	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
32		vex 3500	. . . . . 6 ⊢ 𝑤 ∈ V
33		soex 7629	. . . . . 6 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
34	31, 32, 33	sylancl 588	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐴 ∈ V)
35	18, 34	rabexd 5239	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ∈ V)
36	18	ssrab3 4060	. . . . 5 ⊢ 𝐷 ⊆ 𝐴
37	36	a1i 11	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ⊆ 𝐴)
38		simprr 771	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ≠ ∅)
39		wereu 5554	. . . 4 ⊢ ((𝑤 We 𝐴 ∧ (𝐷 ∈ V ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅)) → ∃!𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣)
40	29, 35, 37, 38, 39	syl13anc 1368	. . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → ∃!𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣)
41		riotacl 7134	. . 3 ⊢ (∃!𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣 → (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣) ∈ 𝐷)
42	40, 41	syl 17	. 2 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣) ∈ 𝐷)
43	28, 42	eqeltrd 2916	1 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃!wreu 3143  {crab 3145  Vcvv 3497   ⊆ wss 3939  ∅c0 4294   class class class wbr 5069   ↦ cmpt 5149   Or wor 5476   We wwe 5516  ran crn 5559   ↾ cres 5560   “ cima 5561  Oncon0 6194  Fun wfun 6352   Fn wfn 6353  ‘cfv 6358  ℩crio 7116  recscrecs 8010

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-wrecs 7950  df-recs 8011

This theorem is referenced by:  zorn2lem2  9922  zorn2lem3  9923  zorn2lem4  9924  zorn2lem5  9925