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Mirrors > Home > MPE Home > Th. List > mptmpoopabovd | Structured version Visualization version GIF version |
Description: The operation value of a function value of a collection of ordered pairs of related elements (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
Ref | Expression |
---|---|
mptmpoopabbrd.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
mptmpoopabbrd.x | ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) |
mptmpoopabbrd.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) |
mptmpoopabbrd.v | ⊢ (𝜑 → {〈𝑓, ℎ〉 ∣ 𝜓} ∈ 𝑉) |
mptmpoopabbrd.r | ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓) |
mptmpoopabovd.m | ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {〈𝑓, ℎ〉 ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)})) |
Ref | Expression |
---|---|
mptmpoopabovd | ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {〈𝑓, ℎ〉 ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptmpoopabbrd.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
2 | mptmpoopabbrd.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) | |
3 | mptmpoopabbrd.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) | |
4 | mptmpoopabbrd.v | . 2 ⊢ (𝜑 → {〈𝑓, ℎ〉 ∣ 𝜓} ∈ 𝑉) | |
5 | mptmpoopabbrd.r | . 2 ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓) | |
6 | oveq12 7165 | . . 3 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑎(𝐶‘𝐺)𝑏) = (𝑋(𝐶‘𝐺)𝑌)) | |
7 | 6 | breqd 5077 | . 2 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ ↔ 𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ)) |
8 | fveq2 6670 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝐶‘𝑔) = (𝐶‘𝐺)) | |
9 | 8 | oveqd 7173 | . . 3 ⊢ (𝑔 = 𝐺 → (𝑎(𝐶‘𝑔)𝑏) = (𝑎(𝐶‘𝐺)𝑏)) |
10 | 9 | breqd 5077 | . 2 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ↔ 𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ)) |
11 | mptmpoopabovd.m | . 2 ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {〈𝑓, ℎ〉 ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)})) | |
12 | 1, 2, 3, 4, 5, 7, 10, 11 | mptmpoopabbrd 7778 | 1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {〈𝑓, ℎ〉 ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 {copab 5128 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: wksonproplem 27486 trlsonfval 27487 pthsonfval 27521 spthson 27522 |
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