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Theorem suplocexprlem2b 7977

Description: Lemma for suplocexpr 7988. 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 5651	. 2 ⊢ (2^nd ‘𝐵) = (2^nd ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩)
3		fo1st 6329	. . . . . 6 ⊢ 1^st :V–onto→V
4		fofun 5569	. . . . . 6 ⊢ (1^st :V–onto→V → Fun 1^st )
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 1^st
6		npex 7736	. . . . . 6 ⊢ P ∈ V
7	6	ssex 4231	. . . . 5 ⊢ (𝐴 ⊆ P → 𝐴 ∈ V)
8		funimaexg 5421	. . . . 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 4542	. . . 4 ⊢ ((1^st “ 𝐴) ∈ V → ∪ (1^st “ 𝐴) ∈ V)
11	9, 10	syl 14	. . 3 ⊢ (𝐴 ⊆ P → ∪ (1^st “ 𝐴) ∈ V)
12		nqex 7626	. . . 4 ⊢ Q ∈ V
13	12	rabex 4239	. . 3 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ∈ V
14		op2ndg 6323	. . 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 2276	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 1398 ∈ wcel 2202 ∃wrex 2512 {crab 2515 Vcvv 2803 ⊆ wss 3201 ⟨cop 3676 ∪ cuni 3898 ∩ cint 3933 class class class wbr 4093 “ cima 4734 Fun wfun 5327 –onto→wfo 5331 ‘cfv 5333 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7543 <_Q cltq 7548 Pcnp 7554

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-1st 6312 df-2nd 6313 df-qs 6751 df-ni 7567 df-nqqs 7611 df-inp 7729

This theorem is referenced by: suplocexprlemmu 7981 suplocexprlemru 7982 suplocexprlemdisj 7983 suplocexprlemloc 7984 suplocexprlemex 7985 suplocexprlemub 7986

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