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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 3475 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3475 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op1std 8003 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (1st ‘𝑝) = 𝑥) |
5 | 4 | fveq2d 6901 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐹‘(1st ‘𝑝)) = (𝐹‘𝑥)) |
6 | 2, 3 | op2ndd 8004 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (2nd ‘𝑝) = 𝑦) |
7 | 6 | fveq2d 6901 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐹‘(2nd ‘𝑝)) = (𝐹‘𝑦)) |
8 | 5, 7 | opeq12d 4882 | . . 3 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → 〈(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))〉 = 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) |
9 | 8 | mpompt 7534 | . 2 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↦ 〈(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))〉) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) |
10 | 1, 9 | eqtr4i 2759 | 1 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ 〈(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 〈cop 4635 ↦ cmpt 5231 × cxp 5676 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1st c1st 7991 2nd c2nd 7992 UnifOncust 24117 Cnucucn 24193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fv 6556 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 |
This theorem is referenced by: ucnima 24199 |
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