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| Mirrors > Home > MPE Home > Th. List > Mathboxes > f1omoALT | Structured version Visualization version GIF version | ||
| Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Use f1omo 48792 without assuming ax-un 7755. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| f1omoALT.1 | ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) |
| Ref | Expression |
|---|---|
| f1omoALT | ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1omoALT.1 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) | |
| 2 | 1 | fveq1d 6908 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) |
| 3 | 1oex 8516 | . . . 4 ⊢ 1o ∈ V | |
| 4 | 3 | fvconstdomi 48791 | . . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o |
| 5 | 2, 4 | eqbrtrdi 5182 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o) |
| 6 | modom2 9281 | . 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o) | |
| 7 | 5, 6 | sylibr 234 | 1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 {csn 4626 class class class wbr 5143 × cxp 5683 ‘cfv 6561 1oc1o 8499 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1o 8506 df-en 8986 df-dom 8987 df-sdom 8988 |
| This theorem is referenced by: (None) |
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