Theorem wemapso 9545

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 4004	. 2 ⊢ (𝐵 ↑m 𝐴) ⊆ (𝐵 ↑m 𝐴)
3		weso 5667	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4	3	adantr 481	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑅 Or 𝐴)
5		simpr 485	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑆 Or 𝐵)
6		vex 3478	. . . . . 6 ⊢ 𝑎 ∈ V
7	6	difexi 5328	. . . . 5 ⊢ (𝑎 ∖ 𝑏) ∈ V
8	7	dmex 7901	. . . 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 5666	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
11	10	ad2antrr 724	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 Fr 𝐴)
12		difss 4131	. . . . 5 ⊢ (𝑎 ∖ 𝑏) ⊆ 𝑎
13		dmss 5902	. . . . 5 ⊢ ((𝑎 ∖ 𝑏) ⊆ 𝑎 → dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎)
14	12, 13	ax-mp 5	. . . 4 ⊢ dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎
15		simprll 777	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ (𝐵 ↑m 𝐴))
16		elmapi 8842	. . . . 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 6737	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ⊆ 𝐴)
19		simprr 771	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ≠ 𝑏)
20	17	ffnd 6718	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 Fn 𝐴)
21		simprlr 778	. . . . . . . 8 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ (𝐵 ↑m 𝐴))
22		elmapi 8842	. . . . . . . 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 6718	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 Fn 𝐴)
25		fndmdifeq0 7045	. . . . . 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 2985	. . . 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 256	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ≠ ∅)
29		fri 5636	. . 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 837	. 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 9544	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑m 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148  {copab 5210   Or wor 5587   Fr wfr 5628   We wwe 5630  dom cdm 5676   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408   ↑_m cmap 8819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821

This theorem is referenced by:  opsrtoslem2  21616  wepwso  41775