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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 5585 | . . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 〈𝐵, 𝐶〉 ∈ (𝑅 × 𝑆)) | |
4 | 1, 2, 3 | syl2anc 584 | . 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 7148 | . . . . . 6 ⊢ (𝐵𝑀𝐶) = (𝑀‘〈𝐵, 𝐶〉) | |
8 | 7 | eqeq2i 2831 | . . . . 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 5124 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘〈𝐵, 𝐶〉))} |
11 | 6, 10 | eqtri 2841 | . 2 ⊢ 𝐺 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑀‘〈𝐵, 𝐶〉))} |
12 | 4, 5, 11 | fnopabco 34879 | 1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 {copab 5119 × cxp 5546 ∘ ccom 5552 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 |
This theorem is referenced by: (None) |
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