Theorem suplocexprlemell 7933

Description: Lemma for suplocexpr 7945. 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 6320	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5561	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1st Fn V
4		ssv 3249	. . . 4 ⊢ 𝐴 ⊆ V
5		fnssres 5445	. . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴)
6	3, 4, 5	mp2an 426	. . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴
7		eluniimadm 5906	. . 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 5046	. . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴)
10	9	unieqi 3903	. . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴)
11	10	eleq2i 2298	. 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴))
12		fvres 5663	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥))
13	12	eleq2d 2301	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥)))
14	13	rexbiia 2547	. 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))

Syntax hints:   ↔ wb 105   ∈ wcel 2202  ∃wrex 2511  Vcvv 2802   ⊆ wss 3200  ∪ cuni 3893   ↾ cres 4727   “ cima 4728   Fn wfn 5321  –onto→wfo 5324  ‘cfv 5326  1^st c1st 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6303

This theorem is referenced by:  suplocexprlemml  7936  suplocexprlemrl  7937  suplocexprlemdisj  7940  suplocexprlemloc  7941  suplocexprlemex  7942  suplocexprlemlub  7944