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Mirrors > Home > ILE Home > Th. List > suplocexprlemell | GIF version |
Description: Lemma for suplocexpr 7557. 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 6063 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5355 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 1st Fn V |
4 | ssv 3124 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | fnssres 5244 | . . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴) | |
6 | 3, 4, 5 | mp2an 423 | . . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴 |
7 | eluniimadm 5674 | . . 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 4860 | . . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴) | |
10 | 9 | unieqi 3754 | . . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴) |
11 | 10 | eleq2i 2207 | . 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴)) |
12 | fvres 5453 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥)) | |
13 | 12 | eleq2d 2210 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥))) |
14 | 13 | rexbiia 2453 | . 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 1481 ∃wrex 2418 Vcvv 2689 ⊆ wss 3076 ∪ cuni 3744 ↾ cres 4549 “ cima 4550 Fn wfn 5126 –onto→wfo 5129 ‘cfv 5131 1st c1st 6044 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 df-1st 6046 |
This theorem is referenced by: suplocexprlemml 7548 suplocexprlemrl 7549 suplocexprlemdisj 7552 suplocexprlemloc 7553 suplocexprlemex 7554 suplocexprlemlub 7556 |
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