Theorem suplocexprlem2b 7716

Description: Lemma for suplocexpr 7727. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)

Hypothesis

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

Assertion

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

Proof of Theorem suplocexprlem2b

Step	Hyp	Ref	Expression
1		suplocexprlem2b.b	. . 3 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
2	1	fveq2i 5520	. 2 ⊢ (2nd ‘𝐵) = (2nd ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
3		fo1st 6161	. . . . . 6 ⊢ 1st :V–onto→V
4		fofun 5441	. . . . . 6 ⊢ (1st :V–onto→V → Fun 1st )
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 1st
6		npex 7475	. . . . . 6 ⊢ P ∈ V
7	6	ssex 4142	. . . . 5 ⊢ (𝐴 ⊆ P → 𝐴 ∈ V)
8		funimaexg 5302	. . . . 5 ⊢ ((Fun 1st ∧ 𝐴 ∈ V) → (1st “ 𝐴) ∈ V)
9	5, 7, 8	sylancr 414	. . . 4 ⊢ (𝐴 ⊆ P → (1st “ 𝐴) ∈ V)
10		uniexg 4441	. . . 4 ⊢ ((1st “ 𝐴) ∈ V → ∪ (1st “ 𝐴) ∈ V)
11	9, 10	syl 14	. . 3 ⊢ (𝐴 ⊆ P → ∪ (1st “ 𝐴) ∈ V)
12		nqex 7365	. . . 4 ⊢ Q ∈ V
13	12	rabex 4149	. . 3 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V
14		op2ndg 6155	. . 3 ⊢ ((∪ (1st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V) → (2nd ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
15	11, 13, 14	sylancl 413	. 2 ⊢ (𝐴 ⊆ P → (2nd ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
16	2, 15	eqtrid 2222	1 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  {crab 2459  Vcvv 2739   ⊆ wss 3131  ⟨cop 3597  ∪ cuni 3811  ∩ cint 3846   class class class wbr 4005   “ cima 4631  Fun wfun 5212  –onto→wfo 5216  ‘cfv 5218  1^st c1st 6142  2^nd c2nd 6143  Qcnq 7282   <_Q cltq 7287  Pcnp 7293

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6144  df-2nd 6145  df-qs 6544  df-ni 7306  df-nqqs 7350  df-inp 7468

This theorem is referenced by:  suplocexprlemmu  7720  suplocexprlemru  7721  suplocexprlemdisj  7722  suplocexprlemloc  7723  suplocexprlemex  7724  suplocexprlemub  7725