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Mirrors > Home > ILE Home > Th. List > suplocexprlemell | GIF version |
Description: Lemma for suplocexpr 7645. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Ref | Expression |
---|---|
suplocexprlemell | ⊢ (𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 6105 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5394 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 1st Fn V |
4 | ssv 3150 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | fnssres 5283 | . . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴) | |
6 | 3, 4, 5 | mp2an 423 | . . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴 |
7 | eluniimadm 5715 | . . 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 4899 | . . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴) | |
10 | 9 | unieqi 3782 | . . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴) |
11 | 10 | eleq2i 2224 | . 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴)) |
12 | fvres 5492 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥)) | |
13 | 12 | eleq2d 2227 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥))) |
14 | 13 | rexbiia 2472 | . 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)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2128 ∃wrex 2436 Vcvv 2712 ⊆ wss 3102 ∪ cuni 3772 ↾ cres 4588 “ cima 4589 Fn wfn 5165 –onto→wfo 5168 ‘cfv 5170 1st c1st 6086 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fo 5176 df-fv 5178 df-1st 6088 |
This theorem is referenced by: suplocexprlemml 7636 suplocexprlemrl 7637 suplocexprlemdisj 7640 suplocexprlemloc 7641 suplocexprlemex 7642 suplocexprlemlub 7644 |
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