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Theorem suplocexprlemell 7675
Description: Lemma for suplocexpr 7687. 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 6136 . . . . 5 1st :V–onto→V
2 fofn 5422 . . . . 5 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . 4 1st Fn V
4 ssv 3169 . . . 4 𝐴 ⊆ V
5 fnssres 5311 . . . 4 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
63, 4, 5mp2an 424 . . 3 (1st𝐴) Fn 𝐴
7 eluniimadm 5744 . . 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 4924 . . . 4 ((1st𝐴) “ 𝐴) = (1st𝐴)
109unieqi 3806 . . 3 ((1st𝐴) “ 𝐴) = (1st𝐴)
1110eleq2i 2237 . 2 (𝐵 ((1st𝐴) “ 𝐴) ↔ 𝐵 (1st𝐴))
12 fvres 5520 . . . 4 (𝑥𝐴 → ((1st𝐴)‘𝑥) = (1st𝑥))
1312eleq2d 2240 . . 3 (𝑥𝐴 → (𝐵 ∈ ((1st𝐴)‘𝑥) ↔ 𝐵 ∈ (1st𝑥)))
1413rexbiia 2485 . 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 2141  wrex 2449  Vcvv 2730  wss 3121   cuni 3796  cres 4613  cima 4614   Fn wfn 5193  ontowfo 5196  cfv 5198  1st c1st 6117
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119
This theorem is referenced by:  suplocexprlemml  7678  suplocexprlemrl  7679  suplocexprlemdisj  7682  suplocexprlemloc  7683  suplocexprlemex  7684  suplocexprlemlub  7686
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