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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 5072 | . . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 〈𝐵, 𝐶〉 ∈ (𝑅 × 𝑆)) | |
4 | 1, 2, 3 | syl2anc 691 | . 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 6552 | . . . . . 6 ⊢ (𝐵𝑀𝐶) = (𝑀‘〈𝐵, 𝐶〉) | |
8 | 7 | eqeq2i 2622 | . . . . 5 ⊢ (𝑦 = (𝐵𝑀𝐶) ↔ 𝑦 = (𝑀‘〈𝐵, 𝐶〉)) |
9 | 8 | anbi2i 726 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘〈𝐵, 𝐶〉))) |
10 | 9 | opabbii 4649 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘〈𝐵, 𝐶〉))} |
11 | 6, 10 | eqtri 2632 | . 2 ⊢ 𝐺 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘〈𝐵, 𝐶〉))} |
12 | 4, 5, 11 | fnopabco 32687 | 1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 {copab 4642 × cxp 5036 ∘ ccom 5042 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 |
This theorem is referenced by: (None) |
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