Theorem wemapso 9529

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 3953	. 2 ⊢ (𝐵 ↑m 𝐴) ⊆ (𝐵 ↑m 𝐴)
3		weso 5642	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4	3	adantr 486	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑅 Or 𝐴)
5		simpr 490	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑆 Or 𝐵)
6		vex 3455	. . . . . 6 ⊢ 𝑎 ∈ V
7	6	difexi 5292	. . . . 5 ⊢ (𝑎 ∖ 𝑏) ∈ V
8	7	dmex 7910	. . . 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 5641	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
11	10	ad2antrr 739	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑅 Fr 𝐴)
12		difss 4083	. . . . 5 ⊢ (𝑎 ∖ 𝑏) ⊆ 𝑎
13		dmss 5884	. . . . 5 ⊢ ((𝑎 ∖ 𝑏) ⊆ 𝑎 → dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎)
14	12, 13	ax-mp 5	. . . 4 ⊢ dom (𝑎 ∖ 𝑏) ⊆ dom 𝑎
15		simprll 791	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ∈ (𝐵 ↑m 𝐴))
16		elmapi 8853	. . . . 5 ⊢ (𝑎 ∈ (𝐵 ↑m 𝐴) → 𝑎:𝐴⟶𝐵)
17	15, 16	syl 18	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎:𝐴⟶𝐵)
18	14, 17	fssdm 6721	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ⊆ 𝐴)
19		simprr 785	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 ≠ 𝑏)
20	17	ffnd 6702	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑎 Fn 𝐴)
21		simprlr 792	. . . . . . . 8 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 ∈ (𝐵 ↑m 𝐴))
22		elmapi 8853	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐵 ↑m 𝐴) → 𝑏:𝐴⟶𝐵)
23	21, 22	syl 18	. . . . . . 7 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏:𝐴⟶𝐵)
24	23	ffnd 6702	. . . . . 6 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → 𝑏 Fn 𝐴)
25		fndmdifeq0 7035	. . . . . 6 ⊢ ((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) → (dom (𝑎 ∖ 𝑏) = ∅ ↔ 𝑎 = 𝑏))
26	20, 24, 25	syl2anc 596	. . . . 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 3000	. . . 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 260	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑎 ≠ 𝑏)) → dom (𝑎 ∖ 𝑏) ≠ ∅)
29		fri 5609	. . 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 852	. 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 9528	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑m 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  {copab 5167   Or wor 5558   Fr wfr 5601   We wwe 5603  dom cdm 5651   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-map 8833

This theorem is used by:  opsrtoslem2  22345  wepwso  44003