Theorem wemapso 8339

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 3185	. 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 3587	. . 3 ⊢ (𝐵 ↑𝑚 𝐴) ⊆ (𝐵 ↑𝑚 𝐴)
4		simp1 1054	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝐴 ∈ V)
5		weso 5029	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
6	5	3ad2ant2 1076	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑅 Or 𝐴)
7		simp3 1056	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑆 Or 𝐵)
8		simpl1 1057	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝐴 ∈ V)
9		difss 3699	. . . . . . 7 ⊢ (𝑎 ∖ 𝑏) ⊆ 𝑎
10		dmss 5245	. . . . . . 7 ⊢ ((𝑎 ∖ 𝑏) ⊆ 𝑎 → dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎
12		simprll 798	. . . . . . . . 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 7765	. . . . . . . . 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 5958	. . . . . . . 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 5904	. . . . . . 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 3616	. . . . 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 4733	. . . 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 1058	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 We 𝐴)
22		wefr 5028	. . . . 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 792	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ≠ 𝑏)
25		simprlr 799	. . . . . . . . 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 7765	. . . . . . . . 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 5958	. . . . . . . 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 6231	. . . . . . 7 ⊢ ((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
31	16, 29, 30	syl2anc 691	. . . . . 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 2826	. . . . 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 246	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ≠ ∅)
34		fri 5000	. . . 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 1319	. . 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 8338	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴))
37	1, 36	syl3an1 1351	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  {copab 4642   Or wor 4958   Fr wfr 4994   We wwe 4996  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746

This theorem is referenced by:  opsrtoslem2  19306  wepwso  36631