Theorem wemapso 8806

Description: Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)

Hypothesis

Ref	Expression
wemapso.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}

Assertion

Ref	Expression
wemapso	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐵(𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wemapso

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3430	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		wemapso.t	. . 3 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
3		ssid 3878	. . 3 ⊢ (𝐵 ↑𝑚 𝐴) ⊆ (𝐵 ↑𝑚 𝐴)
4		simp1 1116	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝐴 ∈ V)
5		weso 5395	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
6	5	3ad2ant2 1114	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑅 Or 𝐴)
7		simp3 1118	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑆 Or 𝐵)
8		simpl1 1171	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝐴 ∈ V)
9		difss 3997	. . . . . . 7 ⊢ (𝑎 ∖ 𝑏) ⊆ 𝑎
10		dmss 5618	. . . . . . 7 ⊢ ((𝑎 ∖ 𝑏) ⊆ 𝑎 → dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎
12		simprll 766	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ (𝐵 ↑𝑚 𝐴))
13		elmapi 8224	. . . . . . 7 ⊢ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) → 𝑎:𝐴⟶𝐵)
14	12, 13	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎:𝐴⟶𝐵)
15	11, 14	fssdm 6358	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ⊆ 𝐴)
16	8, 15	ssexd 5082	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ∈ V)
17		simpl2 1172	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 We 𝐴)
18		wefr 5394	. . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
19	17, 18	syl 17	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 Fr 𝐴)
20		simprr 760	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ≠ 𝑏)
21	14	ffnd 6343	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 Fn 𝐴)
22		simprlr 767	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ (𝐵 ↑𝑚 𝐴))
23		elmapi 8224	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑𝑚 𝐴) → 𝑏:𝐴⟶𝐵)
24	22, 23	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏:𝐴⟶𝐵)
25	24	ffnd 6343	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 Fn 𝐴)
26		fndmdifeq0 6637	. . . . . . 7 ⊢ ((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
27	21, 25, 26	syl2anc 576	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
28	27	necon3bid 3008	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → (dom (𝑎 ∖ 𝑏) ≠ ∅ ↔ 𝑎 ≠ 𝑏))
29	20, 28	mpbird 249	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ≠ ∅)
30		fri 5366	. . . 4 ⊢ (((dom (𝑎 ∖ 𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎 ∖ 𝑏) ⊆ 𝐴 ∧ dom (𝑎 ∖ 𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐)
31	16, 19, 15, 29, 30	syl22anc 826	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐)
32	2, 3, 4, 6, 7, 31	wemapsolem 8805	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴))
33	1, 32	syl3an1 1143	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048   ≠ wne 2964  ∀wral 3085  ∃wrex 3086  Vcvv 3412   ∖ cdif 3825   ⊆ wss 3828  ∅c0 4177   class class class wbr 4927  {copab 4989   Or wor 5322   Fr wfr 5360   We wwe 5362  dom cdm 5404   Fn wfn 6181  ⟶wf 6182  ‘cfv 6186  (class class class)co 6974   ↑_𝑚 cmap 8202

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 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277

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 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-fv 6194  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7498  df-2nd 7499  df-map 8204

This theorem is referenced by:  opsrtoslem2  19972  wepwso  39017