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Theorem suplocexprlemell 7712

Description: Lemma for suplocexpr 7724. 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 6158	. . . . 5 ⊢ 1^st :V–onto→V
2		fofn 5441	. . . . 5 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . . 4 ⊢ 1^st Fn V
4		ssv 3178	. . . 4 ⊢ 𝐴 ⊆ V
5		fnssres 5330	. . . 4 ⊢ ((1^st Fn V ∧ 𝐴 ⊆ V) → (1^st ↾ 𝐴) Fn 𝐴)
6	3, 4, 5	mp2an 426	. . 3 ⊢ (1^st ↾ 𝐴) Fn 𝐴
7		eluniimadm 5766	. . 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 4941	. . . 4 ⊢ ((1^st ↾ 𝐴) “ 𝐴) = (1^st “ 𝐴)
10	9	unieqi 3820	. . 3 ⊢ ∪ ((1^st ↾ 𝐴) “ 𝐴) = ∪ (1^st “ 𝐴)
11	10	eleq2i 2244	. 2 ⊢ (𝐵 ∈ ∪ ((1^st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1^st “ 𝐴))
12		fvres 5540	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1^st ↾ 𝐴)‘𝑥) = (1^st ‘𝑥))
13	12	eleq2d 2247	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1^st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1^st ‘𝑥)))
14	13	rexbiia 2492	. 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

Syntax hints: ↔ wb 105 ∈ wcel 2148 ∃wrex 2456 Vcvv 2738 ⊆ wss 3130 ∪ cuni 3810 ↾ cres 4629 “ cima 4630 Fn wfn 5212 –onto→wfo 5215 ‘cfv 5217 1^st c1st 6139

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-sbc 2964 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fo 5223 df-fv 5225 df-1st 6141

This theorem is referenced by: suplocexprlemml 7715 suplocexprlemrl 7716 suplocexprlemdisj 7719 suplocexprlemloc 7720 suplocexprlemex 7721 suplocexprlemlub 7723

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