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Theorem opsrtoslem1 21608

Description: Lemma for opsrtos 21610. (Contributed by Mario Carneiro, 8-Feb-2015.)

**Hypotheses**
Ref	Expression
opsrso.o	⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
opsrso.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
opsrso.r	⊢ (𝜑 → 𝑅 ∈ Toset)
opsrso.t	⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))
opsrso.w	⊢ (𝜑 → 𝑇 We 𝐼)
opsrtoslem.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
opsrtoslem.b	⊢ 𝐵 = (Base‘𝑆)
opsrtoslem.q	⊢ < = (lt‘𝑅)
opsrtoslem.c	⊢ 𝐶 = (𝑇 <_bag 𝐼)
opsrtoslem.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
opsrtoslem.ps	⊢ (𝜓 ↔ ∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))
opsrtoslem.l	⊢ ≤ = (le‘𝑂)

**Assertion**
Ref	Expression
opsrtoslem1	⊢ (𝜑 → ≤ = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))

Distinct variable groups: 𝑥,𝑦,𝐵 𝑥,𝑤,𝑦,𝑧,𝐶 𝑤,ℎ,𝑥,𝑦,𝑧,𝐼 𝜑,ℎ,𝑤,𝑥,𝑦,𝑧 𝑤,𝐷,𝑥,𝑦,𝑧 𝑤, < ,𝑥,𝑦,𝑧 𝑤,𝑅,𝑥,𝑦,𝑧 𝑤,𝑇,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜓(𝑥,𝑦,𝑧,𝑤,ℎ) 𝐵(𝑧,𝑤,ℎ) 𝐶(ℎ) 𝐷(ℎ) 𝑅(ℎ) 𝑆(𝑥,𝑦,𝑧,𝑤,ℎ) < (ℎ) 𝑇(ℎ) ≤ (𝑥,𝑦,𝑧,𝑤,ℎ) 𝑂(𝑥,𝑦,𝑧,𝑤,ℎ) 𝑉(𝑥,𝑦,𝑧,𝑤,ℎ)

**Proof of Theorem opsrtoslem1**
Step	Hyp	Ref	Expression
1		opsrtoslem.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		opsrso.o	. . 3 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
3		opsrtoslem.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
4		opsrtoslem.q	. . 3 ⊢ < = (lt‘𝑅)
5		opsrtoslem.c	. . 3 ⊢ 𝐶 = (𝑇 <_bag 𝐼)
6		opsrtoslem.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
7		opsrtoslem.l	. . 3 ⊢ ≤ = (le‘𝑂)
8		opsrso.t	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))
9	1, 2, 3, 4, 5, 6, 7, 8	opsrle 21594	. 2 ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})
10		unopab 5230	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓)} ∪ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦))}
11		inopab 5828	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)}) = {⟨𝑥, 𝑦⟩ ∣ (𝜓 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))}
12		df-xp 5682	. . . . . 6 ⊢ (𝐵 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)}
13	12	ineq2i 4209	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)})
14		vex 3479	. . . . . . . . 9 ⊢ 𝑥 ∈ V
15		vex 3479	. . . . . . . . 9 ⊢ 𝑦 ∈ V
16	14, 15	prss 4823	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
17	16	anbi1i 625	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓))
18		ancom 462	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ (𝜓 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
19	17, 18	bitr3i 277	. . . . . 6 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓) ↔ (𝜓 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
20	19	opabbii 5215	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ (𝜓 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))}
21	11, 13, 20	3eqtr4i 2771	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓)}
22		opabresid 6048	. . . . 5 ⊢ ( I ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)}
23		equcom 2022	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
24	23	anbi2i 624	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥))
25		eleq1w 2817	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
26	25	biimpac 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑦) → 𝑦 ∈ 𝐵)
27	26	pm4.71i 561	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑦) ∧ 𝑦 ∈ 𝐵))
28	24, 27	bitr3i 277	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑦) ∧ 𝑦 ∈ 𝐵))
29		an32 645	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝑦) ∧ 𝑦 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = 𝑦))
30	16	anbi1i 625	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦))
31	28, 29, 30	3bitri 297	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦))
32	31	opabbii 5215	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦)}
33	22, 32	eqtri 2761	. . . 4 ⊢ ( I ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦)}
34	21, 33	uneq12i 4161	. . 3 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)) = ({⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓)} ∪ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦)})
35		opsrtoslem.ps	. . . . . . 7 ⊢ (𝜓 ↔ ∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))))
36	35	orbi1i 913	. . . . . 6 ⊢ ((𝜓 ∨ 𝑥 = 𝑦) ↔ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))
37	36	anbi2i 624	. . . . 5 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ (𝜓 ∨ 𝑥 = 𝑦)) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)))
38		andi 1007	. . . . 5 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ (𝜓 ∨ 𝑥 = 𝑦)) ↔ (({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦)))
39	37, 38	bitr3i 277	. . . 4 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)) ↔ (({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦)))
40	39	opabbii 5215	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝜓) ∨ ({𝑥, 𝑦} ⊆ 𝐵 ∧ 𝑥 = 𝑦))}
41	10, 34, 40	3eqtr4ri 2772	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵))
42	9, 41	eqtrdi 2789	1 ⊢ (𝜑 → ≤ = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 {cpr 4630 class class class wbr 5148 {copab 5210 I cid 5573 We wwe 5630 × cxp 5674 ^◡ccnv 5675 ↾ cres 5678 “ cima 5679 ‘cfv 6541 (class class class)co 7406 ↑_m cmap 8817 Fincfn 8936 ℕcn 12209 ℕ₀cn0 12469 Basecbs 17141 lecple 17201 ltcplt 18258 Tosetctos 18366 mPwSer cmps 21449 <_bag cltb 21452 ordPwSer copws 21453

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 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-tr 5266 df-id 5574 df-eprel 5580 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-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-ltxr 11250 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-dec 12675 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ple 17214 df-psr 21454 df-opsr 21458

This theorem is referenced by: opsrtoslem2 21609

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