Theorem suplocexprlemell 8080

Description: Lemma for suplocexpr 8092. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)

Assertion

Ref	Expression
suplocexprlemell	⊢ (𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem suplocexprlemell

Step	Hyp	Ref	Expression
1		fo1st 6391	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5617	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1st Fn V
4		ssv 3270	. . . 4 ⊢ 𝐴 ⊆ V
5		fnssres 5496	. . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴)
6	3, 4, 5	mp2an 430	. . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴
7		eluniimadm 5971	. . 3 ⊢ ((1st ↾ 𝐴) Fn 𝐴 → (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥)))
8	6, 7	ax-mp 5	. 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥))
9		resima 5096	. . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴)
10	9	unieqi 3945	. . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴)
11	10	eleq2i 2305	. 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴))
12		fvres 5719	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥))
13	12	eleq2d 2308	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥)))
14	13	rexbiia 2565	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))
15	8, 11, 14	3bitr3i 210	1 ⊢ (𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))

This proof depends on syntax axioms:   ↔ wb 105   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821   ⊆ wss 3220  ∪ cuni 3935   ↾ cres 4776   “ cima 4777   Fn wfn 5372  –onto→wfo 5375  ‘cfv 5377  1^st c1st 6372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-1st 6374

This theorem is used by:  suplocexprlemml  8083  suplocexprlemrl  8084  suplocexprlemdisj  8087  suplocexprlemloc  8088  suplocexprlemex  8089  suplocexprlemlub  8091