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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  –onto→wfo 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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