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Theorem suplocexprlem2b 7774

Description: Lemma for suplocexpr 7785. 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 5557	. 2 ⊢ (2^nd ‘𝐵) = (2^nd ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩)
3		fo1st 6210	. . . . . 6 ⊢ 1^st :V–onto→V
4		fofun 5477	. . . . . 6 ⊢ (1^st :V–onto→V → Fun 1^st )
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 1^st
6		npex 7533	. . . . . 6 ⊢ P ∈ V
7	6	ssex 4166	. . . . 5 ⊢ (𝐴 ⊆ P → 𝐴 ∈ V)
8		funimaexg 5338	. . . . 5 ⊢ ((Fun 1^st ∧ 𝐴 ∈ V) → (1^st “ 𝐴) ∈ V)
9	5, 7, 8	sylancr 414	. . . 4 ⊢ (𝐴 ⊆ P → (1^st “ 𝐴) ∈ V)
10		uniexg 4470	. . . 4 ⊢ ((1^st “ 𝐴) ∈ V → ∪ (1^st “ 𝐴) ∈ V)
11	9, 10	syl 14	. . 3 ⊢ (𝐴 ⊆ P → ∪ (1^st “ 𝐴) ∈ V)
12		nqex 7423	. . . 4 ⊢ Q ∈ V
13	12	rabex 4173	. . 3 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ∈ V
14		op2ndg 6204	. . 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 413	. 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	eqtrid 2238	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 1364 ∈ wcel 2164 ∃wrex 2473 {crab 2476 Vcvv 2760 ⊆ wss 3153 ⟨cop 3621 ∪ cuni 3835 ∩ cint 3870 class class class wbr 4029 “ cima 4662 Fun wfun 5248 –onto→wfo 5252 ‘cfv 5254 1^st c1st 6191 2^nd c2nd 6192 Qcnq 7340 <_Q cltq 7345 Pcnp 7351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-iinf 4620

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-1st 6193 df-2nd 6194 df-qs 6593 df-ni 7364 df-nqqs 7408 df-inp 7526

This theorem is referenced by: suplocexprlemmu 7778 suplocexprlemru 7779 suplocexprlemdisj 7780 suplocexprlemloc 7781 suplocexprlemex 7782 suplocexprlemub 7783

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