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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opropabco | Structured version Visualization version GIF version |
Description: Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) |
Ref | Expression |
---|---|
opropabco.1 | ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑅) |
opropabco.2 | ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑆) |
opropabco.3 | ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝐵, 𝐶⟩)} |
opropabco.4 | ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} |
Ref | Expression |
---|---|
opropabco | ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opropabco.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑅) | |
2 | opropabco.2 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑆) | |
3 | opelxpi 5715 | . . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆)) | |
4 | 1, 2, 3 | syl2anc 583 | . 2 ⊢ (𝑥 ∈ 𝐴 → ⟨𝐵, 𝐶⟩ ∈ (𝑅 × 𝑆)) |
5 | opropabco.3 | . 2 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝐵, 𝐶⟩)} | |
6 | opropabco.4 | . . 3 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} | |
7 | df-ov 7423 | . . . . . 6 ⊢ (𝐵𝑀𝐶) = (𝑀‘⟨𝐵, 𝐶⟩) | |
8 | 7 | eqeq2i 2741 | . . . . 5 ⊢ (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩)) |
9 | 8 | anbi2i 622 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))) |
10 | 9 | opabbii 5215 | . . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))} |
11 | 6, 10 | eqtri 2756 | . 2 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘⟨𝐵, 𝐶⟩))} |
12 | 4, 5, 11 | fnopabco 37196 | 1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⟨cop 4635 {copab 5210 × cxp 5676 ∘ ccom 5682 Fn wfn 6543 ‘cfv 6548 (class class class)co 7420 |
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 |
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-ne 2938 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-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-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 |
This theorem is referenced by: (None) |
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