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 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem suplocexprlemell

Step	Hyp	Ref	Expression
1		fo1st 6136	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5422	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1st Fn V
4		ssv 3169	. . . 4 ⊢ 𝐴 ⊆ V
5		fnssres 5311	. . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴)
6	3, 4, 5	mp2an 424	. . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴
7		eluniimadm 5744	. . 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 4924	. . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴)
10	9	unieqi 3806	. . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴)
11	10	eleq2i 2237	. 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴))
12		fvres 5520	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥))
13	12	eleq2d 2240	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥)))
14	13	rexbiia 2485	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))
15	8, 11, 14	3bitr3i 209	1 ⊢ (𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))

Syntax hints:   ↔ wb 104   ∈ wcel 2141  ∃wrex 2449  Vcvv 2730   ⊆ wss 3121  ∪ cuni 3796   ↾ cres 4613   “ cima 4614   Fn wfn 5193  –onto→wfo 5196  ‘cfv 5198  1^st 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