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Theorem suplocexprlemell 8087

Description: Lemma for suplocexpr 8099. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)

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

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem suplocexprlemell**
Step	Hyp	Ref	Expression
1		fo1st 6395	. . . . 5 ⊢ 1^st :V–onto→V
2		fofn 5620	. . . . 5 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1^st Fn V
4		ssv 3270	. . . 4 ⊢ 𝐴 ⊆ V
5		fnssres 5499	. . . 4 ⊢ ((1^st Fn V ∧ 𝐴 ⊆ V) → (1^st ↾ 𝐴) Fn 𝐴)
6	3, 4, 5	mp2an 430	. . 3 ⊢ (1^st ↾ 𝐴) Fn 𝐴
7		eluniimadm 5974	. . 3 ⊢ ((1^st ↾ 𝐴) Fn 𝐴 → (𝐵 ∈ ∪ ((1^st ↾ 𝐴) “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ ((1^st ↾ 𝐴)‘𝑥)))
8	6, 7	ax-mp 5	. 2 ⊢ (𝐵 ∈ ∪ ((1^st ↾ 𝐴) “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ ((1^st ↾ 𝐴)‘𝑥))
9		resima 5098	. . . 4 ⊢ ((1^st ↾ 𝐴) “ 𝐴) = (1^st “ 𝐴)
10	9	unieqi 3945	. . 3 ⊢ ∪ ((1^st ↾ 𝐴) “ 𝐴) = ∪ (1^st “ 𝐴)
11	10	eleq2i 2305	. 2 ⊢ (𝐵 ∈ ∪ ((1^st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1^st “ 𝐴))
12		fvres 5722	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1^st ↾ 𝐴)‘𝑥) = (1^st ‘𝑥))
13	12	eleq2d 2308	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1^st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1^st ‘𝑥)))
14	13	rexbiia 2565	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ∈ ((1^st ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1^st ‘𝑥))
15	8, 11, 14	3bitr3i 210	1 ⊢ (𝐵 ∈ ∪ (1^st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1^st ‘𝑥))

Colors of variables: wff set class

This proof depends on syntax axioms: ↔ wb 105 ∈ wcel 2209 ∃wrex 2529 Vcvv 2821 ⊆ wss 3220 ∪ cuni 3935 ↾ cres 4778 “ cima 4779 Fn wfn 5375 –onto→wfo 5378 ‘cfv 5380 1^st c1st 6376

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 4251 ax-pow 4313 ax-pr 4348 ax-un 4580

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 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-fo 5386 df-fv 5388 df-1st 6378

This theorem is used by: suplocexprlemml 8090 suplocexprlemrl 8091 suplocexprlemdisj 8094 suplocexprlemloc 8095 suplocexprlemex 8096 suplocexprlemlub 8098

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