Theorem wemapso 9573

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.) (Revised by AV, 21-Jul-2024.)

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	⊢ ((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)

Proof of Theorem wemapso

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

Step	Hyp	Ref	Expression
1		wemapso.t	. 2 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
2		ssid 3986	. 2 ⊢ (𝐵 ↑m 𝐴) ⊆ (𝐵 ↑m 𝐴)
3		weso 5656	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4	3	adantr 480	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑅 Or 𝐴)
5		simpr 484	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑆 Or 𝐵)
6		vex 3467	. . . . . 6 ⊢ 𝑎 ∈ V
7	6	difexi 5310	. . . . 5 ⊢ (𝑎 ∖ 𝑏) ∈ V
8	7	dmex 7913	. . . 4 ⊢ dom (𝑎 ∖ 𝑏) ∈ V
9	8	a1i 11	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ∈ V)
10		wefr 5655	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
11	10	ad2antrr 726	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 Fr 𝐴)
12		difss 4116	. . . . 5 ⊢ (𝑎 ∖ 𝑏) ⊆ 𝑎
13		dmss 5893	. . . . 5 ⊢ ((𝑎 ∖ 𝑏) ⊆ 𝑎 → dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎)
14	12, 13	ax-mp 5	. . . 4 ⊢ dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎
15		simprll 778	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ (𝐵 ↑m 𝐴))
16		elmapi 8871	. . . . 5 ⊢ (𝑎 ∈ (𝐵 ↑m 𝐴) → 𝑎:𝐴⟶𝐵)
17	15, 16	syl 17	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎:𝐴⟶𝐵)
18	14, 17	fssdm 6735	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ⊆ 𝐴)
19		simprr 772	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ≠ 𝑏)
20	17	ffnd 6717	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 Fn 𝐴)
21		simprlr 779	. . . . . . . 8 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ (𝐵 ↑m 𝐴))
22		elmapi 8871	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐵 ↑m 𝐴) → 𝑏:𝐴⟶𝐵)
23	21, 22	syl 17	. . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏:𝐴⟶𝐵)
24	23	ffnd 6717	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 Fn 𝐴)
25		fndmdifeq0 7044	. . . . . 6 ⊢ ((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
26	20, 24, 25	syl2anc 584	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
27	26	necon3bid 2975	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → (dom (𝑎 ∖ 𝑏) ≠ ∅ ↔ 𝑎 ≠ 𝑏))
28	19, 27	mpbird 257	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ≠ ∅)
29		fri 5622	. . 3 ⊢ (((dom (𝑎 ∖ 𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎 ∖ 𝑏) ⊆ 𝐴 ∧ dom (𝑎 ∖ 𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐)
30	9, 11, 18, 28, 29	syl22anc 838	. 2 ⊢ (((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)
31	1, 2, 4, 5, 30	wemapsolem 9572	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑m 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  Vcvv 3463   ∖ cdif 3928   ⊆ wss 3931  ∅c0 4313   class class class wbr 5123  {copab 5185   Or wor 5571   Fr wfr 5614   We wwe 5616  dom cdm 5665   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7413   ↑_m cmap 8848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-map 8850

This theorem is referenced by:  opsrtoslem2  22029  wepwso  43033