Theorem zorn2lem1 10432

Description: Lemma for zorn2 10442. (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 8344	. . . 4 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
3	2	adantr 481	. . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
4	1	tfr1 8343	. . . . . 6 ⊢ 𝐹 Fn On
5		fnfun 6602	. . . . . 6 ⊢ (𝐹 Fn On → Fun 𝐹)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
7		vex 3449	. . . . 5 ⊢ 𝑥 ∈ V
8		resfunexg 7165	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
9	6, 7, 8	mp2an 690	. . . 4 ⊢ (𝐹 ↾ 𝑥) ∈ V
10		rneq 5891	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
11		df-ima 5646	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
12	10, 11	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
13	12	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑔 ∈ ran 𝑓 ↔ 𝑔 ∈ (𝐹 “ 𝑥)))
14	13	imbi1d 341	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑔 ∈ ran 𝑓 → 𝑔𝑅𝑧) ↔ (𝑔 ∈ (𝐹 “ 𝑥) → 𝑔𝑅𝑧)))
15	14	ralbidv2 3170	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧))
16	15	rabbidv 3415	. . . . . . 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 2801	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → 𝐶 = 𝐷)
20	19	eleq2d 2823	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑢 ∈ 𝐶 ↔ 𝑢 ∈ 𝐷))
21	20	imbi1d 341	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑢 ∈ 𝐶 → ¬ 𝑢𝑤𝑣) ↔ (𝑢 ∈ 𝐷 → ¬ 𝑢𝑤𝑣)))
22	21	ralbidv2 3170	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣 ↔ ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
23	19, 22	riotaeqbidv 7316	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
24		eqid 2736	. . . . 5 ⊢ (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)) = (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))
25		riotaex 7317	. . . . 5 ⊢ (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣) ∈ V
26	23, 24, 25	fvmpt 6948	. . . 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	eqtrdi 2792	. 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 5624	. . . . . . 7 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
31	30	ad2antrl 726	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
32		vex 3449	. . . . . 6 ⊢ 𝑤 ∈ V
33		soex 7858	. . . . . 6 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
34	31, 32, 33	sylancl 586	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐴 ∈ V)
35	18, 34	rabexd 5290	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ∈ V)
36	18	ssrab3 4040	. . . . 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 5629	. . . 4 ⊢ ((𝑤 We 𝐴 ∧ (𝐷 ∈ V ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅)) → ∃!𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣)
40	29, 35, 37, 38, 39	syl13anc 1372	. . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → ∃!𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣)
41		riotacl 7331	. . 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 2838	1 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃!wreu 3351  {crab 3407  Vcvv 3445   ⊆ wss 3910  ∅c0 4282   class class class wbr 5105   ↦ cmpt 5188   Or wor 5544   We wwe 5587  ran crn 5634   ↾ cres 5635   “ cima 5636  Oncon0 6317  Fun wfun 6490   Fn wfn 6491  ‘cfv 6496  ℩crio 7312  recscrecs 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317

This theorem is referenced by:  zorn2lem2  10433  zorn2lem3  10434  zorn2lem4  10435  zorn2lem5  10436