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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 45758 without assuming ax-un 7491. (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 6688 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) |
3 | 1oex 8156 | . . . 4 ⊢ 1o ∈ V | |
4 | 3 | fvconstdomi 45757 | . . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o |
5 | 2, 4 | eqbrtrdi 5079 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o) |
6 | modom2 8811 | . 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 1542 ∈ wcel 2114 ∃*wmo 2539 {csn 4526 class class class wbr 5040 × cxp 5533 ‘cfv 6349 1oc1o 8136 ≼ cdom 8565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7612 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 |
This theorem is referenced by: (None) |
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