Theorem suplocexprlemell 8027

Description: Lemma for suplocexpr 8039. 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 6350	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5591	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1st Fn V
4		ssv 3259	. . . 4 ⊢ 𝐴 ⊆ V
5		fnssres 5470	. . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴)
6	3, 4, 5	mp2an 426	. . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴
7		eluniimadm 5937	. . 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 5070	. . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴)
10	9	unieqi 3923	. . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴)
11	10	eleq2i 2299	. 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴))
12		fvres 5693	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥))
13	12	eleq2d 2302	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥)))
14	13	rexbiia 2557	. 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 2203  ∃wrex 2521  Vcvv 2812   ⊆ wss 3210  ∪ cuni 3913   ↾ cres 4750   “ cima 4751   Fn wfn 5346  –onto→wfo 5349  ‘cfv 5351  1^st c1st 6331

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-1st 6333

This theorem is referenced by:  suplocexprlemml  8030  suplocexprlemrl  8031  suplocexprlemdisj  8034  suplocexprlemloc  8035  suplocexprlemex  8036  suplocexprlemlub  8038