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Theorem suplocexprlemell 7514
Description: Lemma for suplocexpr 7526. 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 6048 . . . . 5 1st :V–onto→V
2 fofn 5342 . . . . 5 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . 4 1st Fn V
4 ssv 3114 . . . 4 𝐴 ⊆ V
5 fnssres 5231 . . . 4 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
63, 4, 5mp2an 422 . . 3 (1st𝐴) Fn 𝐴
7 eluniimadm 5659 . . 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 4847 . . . 4 ((1st𝐴) “ 𝐴) = (1st𝐴)
109unieqi 3741 . . 3 ((1st𝐴) “ 𝐴) = (1st𝐴)
1110eleq2i 2204 . 2 (𝐵 ((1st𝐴) “ 𝐴) ↔ 𝐵 (1st𝐴))
12 fvres 5438 . . . 4 (𝑥𝐴 → ((1st𝐴)‘𝑥) = (1st𝑥))
1312eleq2d 2207 . . 3 (𝑥𝐴 → (𝐵 ∈ ((1st𝐴)‘𝑥) ↔ 𝐵 ∈ (1st𝑥)))
1413rexbiia 2448 . 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 1480  wrex 2415  Vcvv 2681  wss 3066   cuni 3731  cres 4536  cima 4537   Fn wfn 5113  ontowfo 5116  cfv 5118  1st c1st 6029
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031
This theorem is referenced by:  suplocexprlemml  7517  suplocexprlemrl  7518  suplocexprlemdisj  7521  suplocexprlemloc  7522  suplocexprlemex  7523  suplocexprlemlub  7525
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