Theorem zorn2lem1 10565

Description: Lemma for zorn2 10575. (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 8454	. . . 4 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
3	2	adantr 480	. . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
4	1	tfr1 8453	. . . . . 6 ⊢ 𝐹 Fn On
5		fnfun 6679	. . . . . 6 ⊢ (𝐹 Fn On → Fun 𝐹)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
7		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
8		resfunexg 7252	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
9	6, 7, 8	mp2an 691	. . . 4 ⊢ (𝐹 ↾ 𝑥) ∈ V
10		rneq 5961	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
11		df-ima 5713	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
12	10, 11	eqtr4di 2798	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
13	12	eleq2d 2830	. . . . . . . . . 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 3180	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧))
16	15	rabbidv 3451	. . . . . . 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 2805	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → 𝐶 = 𝐷)
20	19	eleq2d 2830	. . . . . . . 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 3180	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣 ↔ ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
23	19, 22	riotaeqbidv 7407	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
24		eqid 2740	. . . . 5 ⊢ (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)) = (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))
25		riotaex 7408	. . . . 5 ⊢ (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣) ∈ V
26	23, 24, 25	fvmpt 7029	. . . 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 2796	. 2 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
29		simprl 770	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝑤 We 𝐴)
30		weso 5691	. . . . . . 7 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
31	30	ad2antrl 727	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
32		vex 3492	. . . . . 6 ⊢ 𝑤 ∈ V
33		soex 7961	. . . . . 6 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
34	31, 32, 33	sylancl 585	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐴 ∈ V)
35	18, 34	rabexd 5358	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ∈ V)
36	18	ssrab3 4105	. . . . 5 ⊢ 𝐷 ⊆ 𝐴
37	36	a1i 11	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ⊆ 𝐴)
38		simprr 772	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ≠ ∅)
39		wereu 5696	. . . 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 7422	. . 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 2844	1 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃!wreu 3386  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166   ↦ cmpt 5249   Or wor 5606   We wwe 5651  ran crn 5701   ↾ cres 5702   “ cima 5703  Oncon0 6395  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  ℩crio 7403  recscrecs 8426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427

This theorem is referenced by:  zorn2lem2  10566  zorn2lem3  10567  zorn2lem4  10568  zorn2lem5  10569