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Mirrors > Home > MPE Home > Th. List > ucnimalem | Structured version Visualization version GIF version |
Description: Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
Ref | Expression |
---|---|
ucnprima.1 | ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) |
ucnprima.2 | ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) |
ucnprima.3 | ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) |
ucnprima.4 | ⊢ (𝜑 → 𝑊 ∈ 𝑉) |
ucnprima.5 | ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) |
Ref | Expression |
---|---|
ucnimalem | ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ 〈(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ucnprima.5 | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op1std 7698 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (1st ‘𝑝) = 𝑥) |
5 | 4 | fveq2d 6673 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐹‘(1st ‘𝑝)) = (𝐹‘𝑥)) |
6 | 2, 3 | op2ndd 7699 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (2nd ‘𝑝) = 𝑦) |
7 | 6 | fveq2d 6673 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐹‘(2nd ‘𝑝)) = (𝐹‘𝑦)) |
8 | 5, 7 | opeq12d 4810 | . . 3 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → 〈(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))〉 = 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) |
9 | 8 | mpompt 7265 | . 2 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↦ 〈(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))〉) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) |
10 | 1, 9 | eqtr4i 2847 | 1 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ 〈(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4572 ↦ cmpt 5145 × cxp 5552 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 1st c1st 7686 2nd c2nd 7687 UnifOncust 22807 Cnucucn 22883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fv 6362 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 |
This theorem is referenced by: ucnima 22889 |
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