Theorem zorn2lem1 9710

Description: Lemma for zorn2 9720. (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 7832	. . . 4 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
3	2	adantr 473	. . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) = ((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹 ↾ 𝑥)))
4	1	tfr1 7831	. . . . . 6 ⊢ 𝐹 Fn On
5		fnfun 6280	. . . . . 6 ⊢ (𝐹 Fn On → Fun 𝐹)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
7		vex 3412	. . . . 5 ⊢ 𝑥 ∈ V
8		resfunexg 6798	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
9	6, 7, 8	mp2an 679	. . . 4 ⊢ (𝐹 ↾ 𝑥) ∈ V
10		rneq 5643	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
11		df-ima 5414	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
12	10, 11	syl6eqr 2826	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
13	12	eleq2d 2845	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑔 ∈ ran 𝑓 ↔ 𝑔 ∈ (𝐹 “ 𝑥)))
14	13	imbi1d 334	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑔 ∈ ran 𝑓 → 𝑔𝑅𝑧) ↔ (𝑔 ∈ (𝐹 “ 𝑥) → 𝑔𝑅𝑧)))
15	14	ralbidv2 3139	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧))
16	15	rabbidv 3397	. . . . . . 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 2833	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → 𝐶 = 𝐷)
20	19	eleq2d 2845	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑢 ∈ 𝐶 ↔ 𝑢 ∈ 𝐷))
21	20	imbi1d 334	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑢 ∈ 𝐶 → ¬ 𝑢𝑤𝑣) ↔ (𝑢 ∈ 𝐷 → ¬ 𝑢𝑤𝑣)))
22	21	ralbidv2 3139	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣 ↔ ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
23	19, 22	riotaeqbidv 6934	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
24		eqid 2772	. . . . 5 ⊢ (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)) = (𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))
25		riotaex 6935	. . . . 5 ⊢ (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣) ∈ V
26	23, 24, 25	fvmpt 6589	. . . 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 2824	. 2 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) = (℩𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣))
29		simprl 758	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝑤 We 𝐴)
30		weso 5392	. . . . . . 7 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
31	30	ad2antrl 715	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
32		vex 3412	. . . . . 6 ⊢ 𝑤 ∈ V
33		soex 7435	. . . . . 6 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
34	31, 32, 33	sylancl 577	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐴 ∈ V)
35	18, 34	rabexd 5086	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ∈ V)
36	18	ssrab3 3941	. . . . 5 ⊢ 𝐷 ⊆ 𝐴
37	36	a1i 11	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ⊆ 𝐴)
38		simprr 760	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → 𝐷 ≠ ∅)
39		wereu 5397	. . . 4 ⊢ ((𝑤 We 𝐴 ∧ (𝐷 ∈ V ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅)) → ∃!𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣)
40	29, 35, 37, 38, 39	syl13anc 1352	. . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → ∃!𝑣 ∈ 𝐷 ∀𝑢 ∈ 𝐷 ¬ 𝑢𝑤𝑣)
41		riotacl 6945	. . 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 2860	1 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ≠ wne 2961  ∀wral 3082  ∃!wreu 3084  {crab 3086  Vcvv 3409   ⊆ wss 3823  ∅c0 4172   class class class wbr 4923   ↦ cmpt 5002   Or wor 5319   We wwe 5359  ran crn 5402   ↾ cres 5403   “ cima 5404  Oncon0 6023  Fun wfun 6176   Fn wfn 6177  ‘cfv 6182  ℩crio 6930  recscrecs 7805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-wrecs 7744  df-recs 7806

This theorem is referenced by:  zorn2lem2  9711  zorn2lem3  9712  zorn2lem4  9713  zorn2lem5  9714