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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 45804 without assuming ax-un 7501. (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 6697 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) |
3 | 1oex 8193 | . . . 4 ⊢ 1o ∈ V | |
4 | 3 | fvconstdomi 45803 | . . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o |
5 | 2, 4 | eqbrtrdi 5078 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o) |
6 | modom2 8856 | . 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o) | |
7 | 5, 6 | sylibr 237 | 1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∃*wmo 2537 {csn 4527 class class class wbr 5039 × cxp 5534 ‘cfv 6358 1oc1o 8173 ≼ cdom 8602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 |
This theorem is referenced by: (None) |
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