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Mirrors > Home > ILE Home > Th. List > suplocexprlemell | GIF version |
Description: Lemma for suplocexpr 7687. 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 6136 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5422 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 1st Fn V |
4 | ssv 3169 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | fnssres 5311 | . . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴) | |
6 | 3, 4, 5 | mp2an 424 | . . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴 |
7 | eluniimadm 5744 | . . 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 4924 | . . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴) | |
10 | 9 | unieqi 3806 | . . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴) |
11 | 10 | eleq2i 2237 | . 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴)) |
12 | fvres 5520 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥)) | |
13 | 12 | eleq2d 2240 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥))) |
14 | 13 | rexbiia 2485 | . 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 2141 ∃wrex 2449 Vcvv 2730 ⊆ wss 3121 ∪ cuni 3796 ↾ cres 4613 “ cima 4614 Fn wfn 5193 –onto→wfo 5196 ‘cfv 5198 1st c1st 6117 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 df-1st 6119 |
This theorem is referenced by: suplocexprlemml 7678 suplocexprlemrl 7679 suplocexprlemdisj 7682 suplocexprlemloc 7683 suplocexprlemex 7684 suplocexprlemlub 7686 |
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