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Theorem suplocexprlem2b 8075

Description: Lemma for suplocexpr 8086. 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 5696	. 2 ⊢ (2^nd ‘𝐵) = (2^nd ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩)
3		fo1st 6385	. . . . . 6 ⊢ 1^st :V–onto→V
4		fofun 5614	. . . . . 6 ⊢ (1^st :V–onto→V → Fun 1^st )
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 1^st
6		npex 7834	. . . . . 6 ⊢ P ∈ V
7	6	ssex 4268	. . . . 5 ⊢ (𝐴 ⊆ P → 𝐴 ∈ V)
8		funimaexg 5463	. . . . 5 ⊢ ((Fun 1^st ∧ 𝐴 ∈ V) → (1^st “ 𝐴) ∈ V)
9	5, 7, 8	sylancr 418	. . . 4 ⊢ (𝐴 ⊆ P → (1^st “ 𝐴) ∈ V)
10		uniexg 4583	. . . 4 ⊢ ((1^st “ 𝐴) ∈ V → ∪ (1^st “ 𝐴) ∈ V)
11	9, 10	syl 14	. . 3 ⊢ (𝐴 ⊆ P → ∪ (1^st “ 𝐴) ∈ V)
12		nqex 7724	. . . 4 ⊢ Q ∈ V
13	12	rabex 4278	. . 3 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ∈ V
14		op2ndg 6379	. . 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 417	. 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 2283	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 1402 ∈ wcel 2209 ∃wrex 2529 {crab 2532 Vcvv 2821 ⊆ wss 3220 ⟨cop 3711 ∪ cuni 3933 ∩ cint 3968 class class class wbr 4128 “ cima 4775 Fun wfun 5369 –onto→wfo 5373 ‘cfv 5375 1^st c1st 6366 2^nd c2nd 6367 Qcnq 7641 <_Q cltq 7646 Pcnp 7652

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-iinf 4733

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-id 4436 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-1st 6368 df-2nd 6369 df-qs 6807 df-ni 7665 df-nqqs 7709 df-inp 7827

This theorem is referenced by: suplocexprlemmu 8079 suplocexprlemru 8080 suplocexprlemdisj 8081 suplocexprlemloc 8082 suplocexprlemex 8083 suplocexprlemub 8084

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