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Theorem suplocexprlemell 7633
Description: Lemma for suplocexpr 7645. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
Assertion
Ref Expression
suplocexprlemell (𝐵 (1st𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (1st𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem suplocexprlemell
StepHypRef Expression
1 fo1st 6105 . . . . 5 1st :V–onto→V
2 fofn 5394 . . . . 5 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . 4 1st Fn V
4 ssv 3150 . . . 4 𝐴 ⊆ V
5 fnssres 5283 . . . 4 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
63, 4, 5mp2an 423 . . 3 (1st𝐴) Fn 𝐴
7 eluniimadm 5715 . . 3 ((1st𝐴) Fn 𝐴 → (𝐵 ((1st𝐴) “ 𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ ((1st𝐴)‘𝑥)))
86, 7ax-mp 5 . 2 (𝐵 ((1st𝐴) “ 𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ ((1st𝐴)‘𝑥))
9 resima 4899 . . . 4 ((1st𝐴) “ 𝐴) = (1st𝐴)
109unieqi 3782 . . 3 ((1st𝐴) “ 𝐴) = (1st𝐴)
1110eleq2i 2224 . 2 (𝐵 ((1st𝐴) “ 𝐴) ↔ 𝐵 (1st𝐴))
12 fvres 5492 . . . 4 (𝑥𝐴 → ((1st𝐴)‘𝑥) = (1st𝑥))
1312eleq2d 2227 . . 3 (𝑥𝐴 → (𝐵 ∈ ((1st𝐴)‘𝑥) ↔ 𝐵 ∈ (1st𝑥)))
1413rexbiia 2472 . 2 (∃𝑥𝐴 𝐵 ∈ ((1st𝐴)‘𝑥) ↔ ∃𝑥𝐴 𝐵 ∈ (1st𝑥))
158, 11, 143bitr3i 209 1 (𝐵 (1st𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (1st𝑥))
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 2128  wrex 2436  Vcvv 2712  wss 3102   cuni 3772  cres 4588  cima 4589   Fn wfn 5165  ontowfo 5168  cfv 5170  1st c1st 6086
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fo 5176  df-fv 5178  df-1st 6088
This theorem is referenced by:  suplocexprlemml  7636  suplocexprlemrl  7637  suplocexprlemdisj  7640  suplocexprlemloc  7641  suplocexprlemex  7642  suplocexprlemlub  7644
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