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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 47798 without assuming ax-un 7722. (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 6887 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) |
3 | 1oex 8477 | . . . 4 ⊢ 1o ∈ V | |
4 | 3 | fvconstdomi 47797 | . . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o |
5 | 2, 4 | eqbrtrdi 5180 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o) |
6 | modom2 9247 | . 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o) | |
7 | 5, 6 | sylibr 233 | 1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃*wmo 2526 {csn 4623 class class class wbr 5141 × cxp 5667 ‘cfv 6537 1oc1o 8460 ≼ cdom 8939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-1o 8467 df-en 8942 df-dom 8943 df-sdom 8944 |
This theorem is referenced by: (None) |
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