Theorem wemapso 8456

Description: Construct lexicographic order on a function space based on a well-ordering of the indexes 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 3212	. 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 3624	. . 3 ⊢ (𝐵 ↑𝑚 𝐴) ⊆ (𝐵 ↑𝑚 𝐴)
4		simp1 1061	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝐴 ∈ V)
5		weso 5105	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
6	5	3ad2ant2 1083	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑅 Or 𝐴)
7		simp3 1063	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑆 Or 𝐵)
8		simpl1 1064	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝐴 ∈ V)
9		difss 3737	. . . . . . 7 ⊢ (𝑎 ∖ 𝑏) ⊆ 𝑎
10		dmss 5323	. . . . . . 7 ⊢ ((𝑎 ∖ 𝑏) ⊆ 𝑎 → dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎
12		simprll 802	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ (𝐵 ↑𝑚 𝐴))
13		elmapi 7879	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) → 𝑎:𝐴⟶𝐵)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎:𝐴⟶𝐵)
15		ffn 6045	. . . . . . . 8 ⊢ (𝑎:𝐴⟶𝐵 → 𝑎 Fn 𝐴)
16	14, 15	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 Fn 𝐴)
17		fndm 5990	. . . . . . 7 ⊢ (𝑎 Fn 𝐴 → dom 𝑎 = 𝐴)
18	16, 17	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom 𝑎 = 𝐴)
19	11, 18	syl5sseq 3653	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ⊆ 𝐴)
20	8, 19	ssexd 4805	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ∈ V)
21		simpl2 1065	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 We 𝐴)
22		wefr 5104	. . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
23	21, 22	syl 17	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 Fr 𝐴)
24		simprr 796	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ≠ 𝑏)
25		simprlr 803	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ (𝐵 ↑𝑚 𝐴))
26		elmapi 7879	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑𝑚 𝐴) → 𝑏:𝐴⟶𝐵)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏:𝐴⟶𝐵)
28		ffn 6045	. . . . . . . 8 ⊢ (𝑏:𝐴⟶𝐵 → 𝑏 Fn 𝐴)
29	27, 28	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 Fn 𝐴)
30		fndmdifeq0 6323	. . . . . . 7 ⊢ ((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
31	16, 29, 30	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
32	31	necon3bid 2838	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → (dom (𝑎 ∖ 𝑏) ≠ ∅ ↔ 𝑎 ≠ 𝑏))
33	24, 32	mpbird 247	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ≠ ∅)
34		fri 5076	. . . 4 ⊢ (((dom (𝑎 ∖ 𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎 ∖ 𝑏) ⊆ 𝐴 ∧ dom (𝑎 ∖ 𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐)
35	20, 23, 19, 33, 34	syl22anc 1327	. . 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)
36	2, 3, 4, 6, 7, 35	wemapsolem 8455	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴))
37	1, 36	syl3an1 1359	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653  {copab 4712   Or wor 5034   Fr wfr 5070   We wwe 5072  dom cdm 5114   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  opsrtoslem2  19485  wepwso  37613