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Mirrors > Home > ILE Home > Th. List > suplocexprlemell | GIF version |
Description: Lemma for suplocexpr 7723. 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 6157 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5440 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 1st Fn V |
4 | ssv 3177 | . . . 4 ⊢ 𝐴 ⊆ V | |
5 | fnssres 5329 | . . . 4 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (1st ↾ 𝐴) Fn 𝐴) | |
6 | 3, 4, 5 | mp2an 426 | . . 3 ⊢ (1st ↾ 𝐴) Fn 𝐴 |
7 | eluniimadm 5765 | . . 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 4940 | . . . 4 ⊢ ((1st ↾ 𝐴) “ 𝐴) = (1st “ 𝐴) | |
10 | 9 | unieqi 3819 | . . 3 ⊢ ∪ ((1st ↾ 𝐴) “ 𝐴) = ∪ (1st “ 𝐴) |
11 | 10 | eleq2i 2244 | . 2 ⊢ (𝐵 ∈ ∪ ((1st ↾ 𝐴) “ 𝐴) ↔ 𝐵 ∈ ∪ (1st “ 𝐴)) |
12 | fvres 5539 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((1st ↾ 𝐴)‘𝑥) = (1st ‘𝑥)) | |
13 | 12 | eleq2d 2247 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ 𝐵 ∈ (1st ‘𝑥))) |
14 | 13 | rexbiia 2492 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ∈ ((1st ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥)) |
15 | 8, 11, 14 | 3bitr3i 210 | 1 ⊢ (𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 ∃wrex 2456 Vcvv 2737 ⊆ wss 3129 ∪ cuni 3809 ↾ cres 4628 “ cima 4629 Fn wfn 5211 –onto→wfo 5214 ‘cfv 5216 1st c1st 6138 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
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 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 df-fv 5224 df-1st 6140 |
This theorem is referenced by: suplocexprlemml 7714 suplocexprlemrl 7715 suplocexprlemdisj 7718 suplocexprlemloc 7719 suplocexprlemex 7720 suplocexprlemlub 7722 |
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