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Theorem suplocexprlemell 7711
Description: Lemma for suplocexpr 7723. 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 6157 . . . . 5 1st :V–onto→V
2 fofn 5440 . . . . 5 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . 4 1st Fn V
4 ssv 3177 . . . 4 𝐴 ⊆ V
5 fnssres 5329 . . . 4 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
63, 4, 5mp2an 426 . . 3 (1st𝐴) Fn 𝐴
7 eluniimadm 5765 . . 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 4940 . . . 4 ((1st𝐴) “ 𝐴) = (1st𝐴)
109unieqi 3819 . . 3 ((1st𝐴) “ 𝐴) = (1st𝐴)
1110eleq2i 2244 . 2 (𝐵 ((1st𝐴) “ 𝐴) ↔ 𝐵 (1st𝐴))
12 fvres 5539 . . . 4 (𝑥𝐴 → ((1st𝐴)‘𝑥) = (1st𝑥))
1312eleq2d 2247 . . 3 (𝑥𝐴 → (𝐵 ∈ ((1st𝐴)‘𝑥) ↔ 𝐵 ∈ (1st𝑥)))
1413rexbiia 2492 . 2 (∃𝑥𝐴 𝐵 ∈ ((1st𝐴)‘𝑥) ↔ ∃𝑥𝐴 𝐵 ∈ (1st𝑥))
158, 11, 143bitr3i 210 1 (𝐵 (1st𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (1st𝑥))
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2148  wrex 2456  Vcvv 2737  wss 3129   cuni 3809  cres 4628  cima 4629   Fn wfn 5211  ontowfo 5214  cfv 5216  1st c1st 6138
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-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-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
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-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-1st 6140
This theorem is referenced by:  suplocexprlemml  7714  suplocexprlemrl  7715  suplocexprlemdisj  7718  suplocexprlemloc  7719  suplocexprlemex  7720  suplocexprlemlub  7722
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