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Theorem suplocexprlem2b 7546

Description: Lemma for suplocexpr 7557. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)

**Hypothesis**
Ref	Expression
suplocexprlem2b.b	⊢ 𝐵 = ⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩

**Assertion**
Ref	Expression
suplocexprlem2b	⊢ (𝐴 ⊆ P → (2^nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢})

**Proof of Theorem suplocexprlem2b**
Step	Hyp	Ref	Expression
1		suplocexprlem2b.b	. . 3 ⊢ 𝐵 = ⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩
2	1	fveq2i 5432	. 2 ⊢ (2^nd ‘𝐵) = (2^nd ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩)
3		fo1st 6063	. . . . . 6 ⊢ 1^st :V–onto→V
4		fofun 5354	. . . . . 6 ⊢ (1^st :V–onto→V → Fun 1^st )
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 1^st
6		npex 7305	. . . . . 6 ⊢ P ∈ V
7	6	ssex 4073	. . . . 5 ⊢ (𝐴 ⊆ P → 𝐴 ∈ V)
8		funimaexg 5215	. . . . 5 ⊢ ((Fun 1^st ∧ 𝐴 ∈ V) → (1^st “ 𝐴) ∈ V)
9	5, 7, 8	sylancr 411	. . . 4 ⊢ (𝐴 ⊆ P → (1^st “ 𝐴) ∈ V)
10		uniexg 4369	. . . 4 ⊢ ((1^st “ 𝐴) ∈ V → ∪ (1^st “ 𝐴) ∈ V)
11	9, 10	syl 14	. . 3 ⊢ (𝐴 ⊆ P → ∪ (1^st “ 𝐴) ∈ V)
12		nqex 7195	. . . 4 ⊢ Q ∈ V
13	12	rabex 4080	. . 3 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ∈ V
14		op2ndg 6057	. . 3 ⊢ ((∪ (1^st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ∈ V) → (2^nd ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢})
15	11, 13, 14	sylancl 410	. 2 ⊢ (𝐴 ⊆ P → (2^nd ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢})
16	2, 15	syl5eq 2185	1 ⊢ (𝐴 ⊆ P → (2^nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢})

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 {crab 2421 Vcvv 2689 ⊆ wss 3076 ⟨cop 3535 ∪ cuni 3744 ∩ cint 3779 class class class wbr 3937 “ cima 4550 Fun wfun 5125 –onto→wfo 5129 ‘cfv 5131 1^st c1st 6044 2^nd c2nd 6045 Qcnq 7112 <_Q cltq 7117 Pcnp 7123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1st 6046 df-2nd 6047 df-qs 6443 df-ni 7136 df-nqqs 7180 df-inp 7298

This theorem is referenced by: suplocexprlemmu 7550 suplocexprlemru 7551 suplocexprlemdisj 7552 suplocexprlemloc 7553 suplocexprlemex 7554 suplocexprlemub 7555

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